Queueing Networks and Markov Chains : Modeling and Performance Evaluation With Computer Science Applications

Queueing Networks and Markov Chains

Queueing Networks and Markov Chains Modeling and Performance Evaluation with Computer Science Applications

Gunter Hermann

de Meer,

Belch,

Stefan

and Kishor

Greiner, S. Trivedi

A Wiley-Interscience

JOHN WILEY New York / Chichester

/ Weinheim

Publication

& SONS, INC.

/ Brisbane

/ Singapore

/ Toronto

Copyright  1998 by John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic or mechanical, including uploading, downloading, printing, decompiling, recording or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: [email protected]. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional person should be sought. ISBN 0-471-20058-1. This title is also available in print as ISBN 0-471-19366-6 For more information about Wiley products, visit our web site at www.Wiley.com. Library of Congress Cataloging in Publication Data: Queueing networks and Markov chains: modeling and performance evaluation with computer science applications / Gunter Bolch . . . [et al.]. p. cm. “A Wiley-Interscience publication.” Includes bibliographical references and index. ISBN 0-471-19366-6 1. Electronic digital computers — Evaluation. 2. Markov processes. 3. Queuing theory. I. Bolch, Gunter. QA76.9.E94Q48 1998 97-11959 004.20 40 01519233 — dc21 CIP Printed in the United States of America. 10 9 8 7 6 5 4 3 2

Contents

Preface 1 Introduction 1.1 Motivation 1.2 Basics of Probability and Statistics 1.2.1 Random Variables 1.2.1.1 Discrete Random Variables 1.2. 1.2 Continuous Random Variables 1.2.2 Multiple Random Variables 1.2.2.1 Independence 1.2.2.2 Conditional Probability 1.2.2.3 Important Relations 1.2.2.4 The Central Limit Theorem 1.2.3 Transforms 1.2.3.1 z- Transform 1.2.3.2 Laplace Transform 1.2.4 Parameter Estimation 1.2.4.1 Method of Moments 1.2.4.2 Maximum-Likelihood Estimation 1.2.4.3 Confidence Intervals 1.2.5 Order Statistics

xv 1 1 5 5 5 7 19 20 22 23 24 25 25 26 27 28 29 29 30

vi

CONTENTS

1.2.6

Distribution

of Sums

2

Markov Chains 2.1 Markov Processes 2.1. I Stochastic and Markov Processes 2, I .2 Markov Chains 2. I .2.1 Discrete- Time Markov Chains 2.1.2.2 Continuous- Time Markov Chains 2.1.2.3 Recapitulation 2.2 The Modeling Process 2.2.1 Modeling Life-cycle Phases 2.2.2 Performance Measures 2.2.2.1 A Simple Example 2.2.2.2 Markov Reward Models 2.2.2.3 A Case Study 2.2.3 Generation Methods 2.2.3.1 Petri Nets 2.2.3.2 Generalized Stochastic Petri Nets 2.2.3.3 Stochastic Reward Nets 2.2.3.4 GSPN/SRN Analysis 2.2.3.5 A Larger Example

3

Steady-State Solutions of Markov Chains 3.1 Symbolic Solution: Birth-Death Process 3.2 Hessenberg Matrix: Non-Markovian Queues 3.2.1 Non-Exponential Service Times 3.2.2 Server with Vacations 3.2.2.1 Polling Systems 3.2.2.2 Analysis 3.3 Numerical Solution: Direct Methods 3.3.1 Gaussian Elimination 3.3.2 The Grassmann Algorithm 3.4 Numerical Solution: Iterative Methods 3.4.1 Convergence of Iterative Methods 3.4.2 Power Method 3.4.3 Jacobi’s Method 3.4.4 Gauss-Seidel Method 3.4.5 The Method of Successive Over-Relaxation

31 35 35 35 37 37 49 55 56 56 61 61 64 70 80 81 86 87 91 98 103 105 106 107 112 113 114 118 119 125 132 132 133 136 139 140

CONTENTS

3.5

The Relaxation Parameter 3.4.5.1 3.4.5.2 The Test of Convergence 3.4.5.3 Overflow and Underflow 3.4.5.4 The Algorithm Comparison of Numerical Solution Methods 3.5.1 Case Studies

vii 141 143 143

144 144 146

4

Steady-State Aggregation/ Disaggregation Methods 4.1 Courtois’s Approximate Method 4.1.1 Decomposition 4.1.2 Applicability 4.1.3 Analysis of the Substructures 4.1.4 Aggregation and Unconditioning 4.1.5 The Algorithm 4.2 Takahashi’s Iterative Method 4.2.1 The Fundamental Equations 4.2.2 Applicability 4.2.3 The Algorithm 4.2.4 Application 4.2.4.1 Aggregation 4.2.4.2 Disaggregation 4.2.5 Final Remarks

153 153 154 160 162 163 165 166 167 169 170 170 172 173 174

5

Transient Solution of Markov Chains 5. I Transient Analysis Using Exact Methods 5.1.1 A Pure Birth Process 5.1.2 A Two-State CTMC 5.1.3 Solution Using Laplace Transforms 5.1.4 Numerical Solution Using Uniformixation 5.1.4.1 The Instantaneous Case 5.1.4.2 Stiffness Tolerant Uniformization 5.1.4.3 The Cumulative Case 5.1.5 Other Numerical Methods 5.1.5.1 Ordinary Differential Equations 5.1.5.2 Weak Lumpability 5.2 Aggregation of Stiff Markov Chains 5.2.1 Outline and Basic Definitions 5.2.2 Aggregation of Fast Recurrent Subsets

177 178 178 181

184 184

184 186 187 189 189 189 190 191 192

...

VIII

CONTENTS

52.3 5.2.4 5.2.5

5.2.6 5.2.7

Aggregation of Fast Transient Subsets Aggregation of Initial State Probabilities Disaggregations 5.2.5.1 Fast Transient States 5.2.5.2 Fast Recurrent States The Algorithm An Example: Server Breakdown and Repair

195 196 197 197 197 198 200

6 Single Station Queueing Systems 6.1 Kendall’s Notation 6.2 Performance Measures 6.3 The M/M/l System 6.4 The M/M/m System 6.5 The M/M/m System 6.6 The M/M/l/K Finite Capacity System 6.7 Machine Repairman Model 6.8 Closed Tandem Network 6.9 The M/G/l System 6.10 The GI/M/l System 6.11 The GI,M/m System 6.12 The GI/G/l System 6.13 The M/G/m System 6.14 The GI/G/m System 6.15 Priority Queueing 6.15.1 System without Preemption 6.15.2 Conservation Laws 6.15.3 System with Preemption 6.15.4 System with Time-Dependent Priorities 6.16 The Asymmetric System 6.16.1 Approximate Analysis 6.16.2 Exact Analysis 6.16.2.1 Analysis of M/M/m Loss Systems 6.16.2.2 Extending to Non-Lossy System 6.16.3 Exact Analysis of an Asymmetric M/M/2 System 6.17 Systems with Batch Service

209 210 212 214 217 218 219 221 222 223 225 228 229 230 233 237 239 242 242 243 248 248 249 250 251

7

263

Queueing Networks

253 258

CONTENTS

ix

Definitions and Notation 7.1.1 Single Class Networks 7.1.2 Multiclass Networks Performance Measures 7.2.1 Single Class Networks 7.2.2 Multiclass Networks Product-Form Queueing Networks 7.3.1 Global Balance 7.3.2 Local Balance 7.3.3 Product-Form 7.3.4 Jackson Networks 7.3.5 Gordon/Newell Networks 7.3.6 BCMP Networks 7.3.6.1 The Concept of Chains 7.3.6.2 BCMP Theorem

265 265 267 268 268 271 273

Algorithms for Product-Form Networks 8.1 The Convolution Algorithm 8.1 .l Single Class Closed Networks 8. I. 2 Multiclass Closed Networks 8.2 The Mean Value Analysis 8.2.~ Single Class Closed Networks 8.2.2 Multiclass Closed Networks 8.2.3 Mixed Networks 8.2.4 Networks with Load-Dependent Service 8.2.4.1 Closed Networks 8.2.4.2 Mixed Networks 8.3 The RECAL Method 8.4 Flow Equivalent Server Method 8.4.1 FES Method for a Single Node 8.4.2 FES Method for Multiple Nodes 8.5 Summary

311 313 313 320 326 327 335

9 Approximation Algorithms for Product-Form Networks 9.1 Approximations Based on the MVA 9.1.1 Bard-Schweitzer Approximation 9.1.2 Self- Correcting Approximation Technique 9.1.2.1 Single Server Nodes

379 380 380 385 385

7.1

7.2

7.3

8

274 277 283

284 288 295 296 300

342 347 348 352 360 368 368 373 376

X

CONTENTS

9.2

9.3

9.4

9.5 9.6

9.1.2.2 Multiple Server Nodes 9.1.2.3 Extended SCAT Algorithm Summation Method 9.2.1 Single Class Networks 9.2.2 Multiclass Networks Bottapprox Method 9.3.1 Initial Value of X 9.3.2 Single Class Networks 9.3.3 Multiclass Networks Bounds Analysis 9.4.1 Asymptotic Bounds Analysis 9.4.2 Balanced Job Bounds Analysis Networks with Variabilities in Workload Summary

10 Algorithms for Non-Product-Form Networks 10.1 Non-Exponential Distributions 10.1.1 Diffusion Approximation 10.1.1.1 Open Networks 10.1.1.2 Closed Networks 10.1.2 Maximum Entropy Method 10.1.2.1 Open Networks 10.1.2.2 Closed Networks 10.1.3 Decomposition for Open Networks 10.1.4 Methods for Closed Networks 10.1.4.1 Robustness for Closed Networks 10.1.4-Z Marie’s Method 10.1.4.3 Extended SUM and BOTT 10.1.5 Closing Method for Mixed Networks 10.2 Different Service Times at FCFS Nodes 10.3 Priority Networks 10.3.1 Extended MVA (PRIOMVA) 10.3.2 The Method of Shadow Server 10.3.2.1 The Original Shadow Technique 10.3.2.2 Extensions of the Shadow Technique 10.3.3 Extended SUM 10.4 Simultaneous Resource Possession 10.4.1 Memory Constraints

389 393 397

400 403 405 405

405 408 410 411 414 418 420 421 423 423 424 427 430 431 433 439

448 448 452 463 464 469 470 470 478 478

480 494 497 498

CONTENTS

10.5 10.6

IO. 7

10.8

10.4.2 I/O Subsystems 10.4.3 Method of Surrogate Delays 10.4.4 Serialization Programs with Internal Concurrency Parallel Processing 10.6.1 Asynchronous Tasks 10.6.2 Fork Join Systems 10.6.2.1 Modeling 10.6.2.2 Performance Measures 10.6.2.3 Analysis Networks with Asymmetric Nodes 10.7.1 Closed Networks 10.7.1.1 Asymmetric SUM (ASUM) 10.7.1.2 Asymmetric MVA (AMVA) 10.7.1.3 Asymmetric SCAT (ASCAT) 10.7.2 Open Networks Networks with Blocking 10.8. I Different Blocking Types 10.8.2 Solution for Networks with Two Nodes

xi

501 504 505 506 507 507 517 518 519 521 533 534 534 535 536 537 547 548 549

11 Optimization 11.1 Optimization Problems and Cost Functions 11.2 Optimization based on the Summation Method 11.2.1 Maximization of the Throughput 11.2.2 Minimization of Cost 11.2.3 Minimization of the Response Time 11.3 Optimization based on the Convolution Algorithm 11.3.1 Maximization of the Throughput 11.3.2 Optimal Design of Storage Hierarchies

557 558 559 560 561 562 564 564 566

12 Performance Analysis 12.1 PEPSY 12.1.1 Structure 12.1.1.1 12.1.1.2 12.1.1.3 12.1.2 Different 12.1.3 Example

571 572 573 573 573 574 574 575

Tools of PEPSY

The Input File The Output File Control Files Programs in PEPSY of using PEPSY

xii

CONTENTS

12.1.4 Graphical User Interface XPEPSY 12.2 SPNP 12.2.1 SPNP Features 12.2.2 The CSPL Language 12.2.3 iSPN 12.3 MOSES 12.3.1 The Model Description Language MOSEL 12.3.2 Examples 12.3.2.1 Central-Server Model 12.3.2.2 Fault Tolerant Multiprocessor System 12.4 SHARPE 12.4.1 Central-Server Queueing Network 12.4.2 M/M/m/K System 12.4.3 M/M/l/K System with Server Failure and Repair 12.5 Characteristics of Some Tools 13 Applications 13.1 Case Studies of Queueing Networks 13.1.1 Multiprocessor Systems 13.1.1.1 Tightly Coupled Systems 13.1.1.2 Loosely Coupled Systems 13.1.2 Client Server Systems 13.1.3 Communication Systems 13.1.3.1 Description of the System 13.1.3.2 Queueing Network Model 13.1.3.3 Model Parameters 13.1.3.4 Results 13.1.4 UNIX Kernel 13.1.4.1 Model of the UNIX Kernel 13.1.4.2 Analysis 13.1.5 Flexible Production Systems 13.1.5.1 An Open Network Model 13.1.5.2 A Closed Network Model 13.2 Case Studies of Markov Chains 13.2.1 Wafer Production System 13.2.2 Polling Systems 13.2.3 Client Server Systems

577 579 579 580 585 588 588 590 590 591 594 594 596 597 600 603 603 603 603 606 607 608 608 611 612 617 620 621

624 630 630 634 638 639

641 645

CONTENTS

13.24

ISDN Channel 13.2.4.1 The Baseline Model 13.2.4.2 Markov Modulated Poisson Processes 13.2.5 ATM Network under Overload i3.3 Case Studies of Hierarchical Models 13.3.1 A Multiprocessor with Different Cache Strategies 13.3.2 Performability of a Multiprocessor System

. .. XIII

650 650 653 658 665 666 674

Glossary

679

Bibliography

691

Index

719

Preface Queuemg networks and Markov chains are commonly used for the performance and reliability evaluation of computer, communication, and manufacturing systems. Although there are quite a few books on the individual topics of queueing networks and Markov chains, we have found none that covers both of these topics. The purpose of this book, therefore, is to offer a detailed treatment of queueing systems, queueing networks, and continuous and discrete-time Markov chains. In addition to introducing the basics of these subjects, we have endeavored to: l

Provide some in-depth numerical solution algorithms.

0 Incorporate a rich set of examples that demonstrate the application of the different paradigms and corresponding algorithms. l

Discuss stochastic Petri nets as a high-level description language, thereby facilitating automatic generation and solution of voluminous Markov chains.

l

Treat in some detail approximation els.

l

Describe and apply four software packages throughout the text.

l

Provide problems as exercises.

met hods that will handle large mod-

This book easily lends itself to a course on performance evaluation in the computer science and computer engineering curricula. It can also be used for a course on stochastic models in mathematics, operations research and industrial engineering departments. Because it incorporates a rich and comprehensive set of numerical solution methods comparatively presented, the text may also

xvi

PREFACE

well serve practitioners in various fields of applications as a reference book for algorithms. With sincere appreciation to our friends, colleagues, and students who so ably and patiently supported our manuscript project, we wish to publicly acknowledge: l

J&g Barner and Stephan Kijsters for their painstaking work in keying the text and in laying out the figures and plots.

l

Peter Bazan, who assisted both with the programming of many examples and comprehensive proofreading.

l

Hana Sevcikova, who lent a hand in solving many of the examples and contributed with proofreading.

l

Janos Sztrik, for his comprehensive proofreading.

l

Doris Ehrenreich, who wrote the first version of the section on communication systems.

l

Markus Decker, who prepared the first draft of the mixed queueing networks section.

l

Those who read parts of the manuscript and provided many useful comment s, including: Khalid Begain, Oliver Diisterhoft, Ricardo Fricks, Swapna Gokhale, Thomas Hahn, Christophe Hirel, Graham Horton, Steve Hunter, Demetres Kouvatsos, Yue Ma, Raymond Marie, Varsha Mainkar, Victor Nicola, Cheul Woo Ro, Helena Szczerbicka, Lorrie Tomek, Bernd Wolfinger, Katinka Wolter, Martin Zaddach, and Henry Zang.

Gunter Belch and Stefan Greiner are grateful to Fridolin Hofmann, and Hermann de Meer is grateful to Bernd Wolfinger, for their support in providing the necessary freedom from distracting obligations. Thanks are also due to Teubner B.G. Publishing house for allowing us to borrow sections from the book entitled Leistungsbewertung von Rechensystemen (originally in German) by one of the coauthors, Gunter Belch. In the present book, these sections are integrated in Chapters 1 and 7 through 10. We also thank Andrew Smith, Lisa Van Horn, and Mary Lynn of John Wiley & Sons for their patience and encouragement. The financial support from the SFB (Collaborative Research Centre) 182: “Multiprocessor and Network Configurations” of the DFG (Deutsche Forschungsgemeinschaft) is acknowledged. Finally, a web page has been set up for further information regarding the book. The URL is http://www4.cs.fau.de/QNMC/ GUNTER BOLCH, STEFAN GREINER,

HERMANN

DE MEER, KISHOR

S. TRIVEDI

Queueing Networks and Markov Chains

Introduction

1.1

MOTIVATION

Information processing system designers need methods for the quantification of system design factors such as performance and reliability. Modern computerr communicationI’ and production line systems process complex workloads with random service demands. Probabilistic and statistical methods are commonly employed for the purpose of performance and reliability evaluation. The purpose of this book is to explore major probabilistic modeling techniques for the performance analysis of information processing systems. Statistical methods are also of great importance but we refer the reader to other sources [JainSlI’Triv82] for this topic. Although we concentrate on performance analysisI’we occasionally consider reliabilityI”availabilityI’and combined performance and reliability analysis. Performance measures that are commonly of interest include throughputI’resource utilizationrloss probabili@and delay (or response time). The most direct method for performance evaluation is based on actual measurement of the system under study. HoweverI’during the design phaser the system is not available for such experimentsrand yet performance of a given design needs to be predicted to verify that it meets design requirements and to carry out necessary trade-offs. HenceI’abstract models are necessary for performance prediction of designs. The most popular models are based on discrete-event simulation (DES). DES can be applied to almost all problems of interestrand system details to the desired degree can be captured in such simulation models. F’urthermoreI’many software packages are available that facilitate the construction and execution of DES models. 1

2

INTRODUCTION

The principal drawback of DES models, however, is the time taken to run such models for large, realistic systems particularly when results with high accuracy (i.e., narrow confidence intervals) are desired. A cost-effective alternative to DES models, analytic models can provide relatively quick answers to “what if” questions and can provide more insight into the system being studied. However, analytic models are often plagued by unrealistic assumptions that need to be made in order to make them tractable. Recent advances in stochastic models and numerical solution techniques, availability of software packages, and easy access to workstations with large computational capabilities have extended the capabilities of analytic models to more complex systems. Analytical models can be broadly classified into state space models and non-state space models. Most commonly used state space models are Markov chains. First introduced by A. A. Markov in 1907, Markov chains have been in use in performance analysis since around 1950. In the past decade, considerable advances have been made in the numerical solution techniques, methods of automated state space generation, and the availability of software packages. These advances have resulted in extensive use of Markov chains in performance and reliability analysis. A Markov chain consists of a set of states and a set of labeled transitions between the states. A state of the Markov chain can model various conditions of interest in the system being studied. These could be the number of jobs of various types waiting to use each resource, the number of resources of each type that have failed, the number of concurrent tasks of a given job being executed, and so on. After a sojourn in a state, the Markov chain will make a transition to another state. Such transitions are labeled either with probabilities of transition (in case of discrete-time Markov chains) or rates of transition (in case of continuous-time Markov chains). Long run (steady-state) dynamics of Markov chains can be studied using a system of linear equations with one equation for each state. Transient (or time dependent) behavior of a continuous-time Markov chain gives rise to a system of first-order, linear, ordinary differential equations. Solution of these equations results in state probabilities of the Markov chain from which desired performance measures can be easily obtained. The number of states in a Markov chain of a complex system can become very large and, hence, automated generat ion and efficient numerical solution met hods for underlying equations are desired. A number of concise notations (based on queueing networks and stochastic Petri nets) have evolved, and software packages that automatically generate the underlying state space of the Markov chain are now available. These packages also carry out efficient solution of steady-state and transient behavior of Markov chains. In spite of these advances, there is a continuing need to be able to deal with larger Markov chains and much research is being devoted to this topic. If the Markov chain has nice structure, it is often possible to avoid the generation and solution of the underlying (large) state space. For a class of queueing networks, known as product-form queueing networks (PFQN),

MOTIVATION

3

it is possible to derive steady-state performance measures without resorting to the underlying state space. Such models are therefore called non-state space models. Other examples of non-state space models are directed acyclic task precedence graphs [SaTr87] and fault-trees [STP96]. Other examples of methods exploiting Markov chains with “nice” structure are matrix-geometric methods [Neut81]. We do not discuss these model types in this book due to space limitations. Relatively large PFQN can be solved by means of a small number of simpler equations. However, practical queueing networks can often get so large that approximate methods are needed to solve such PFQN. Furthermore, many practical queueing networks (so-called non-product-form queueing networks, NPFQN) do not satisfy restrictions implied by product-form. In such cases, it is often possible to obtain accurate approximations using variations of algorithms used for PFQNs. Other approximation techniques using hierarchical and fixed-point iterative methods are also used. The flowchart shown in Fig. 1.1 gives the organization of this book. The rest of this chapter, Section 1.2, covers the basics of probability and statistics. In Chapter 2, Markov chains basics are presented together with generation methods for them. Exact steady-state solution techniques for Markov chains are given in Chapter 3 and their aggregation/disaggregation counterpart in Chapter 4. These aggregation/disaggregation solution techniques are useful for practical Markov chain models with very large state spaces. Transient solution techniques for Markov chajns are introduced in Chapter 5. Chapter 6 deals with the description and computation of performance measures for single-station queueing systems in steady state. A general description of queueing networks is given in Chapter 7. Exact solution methods for PFQN are described in detail in Chapter 8 while approximate solution techniques for PFQN are described in Chapter 9. Solution algorithms for different types of NPFQN ( sue h as networks with priorities, non-exponential service times, blocking, or parallel processing) are presented in Chapter 10. The solution algorithms introduced in this book can also be used for optimization problems as described in Chapter 11. For the practical use of modeling techniques described in this book, software packages (tools) are needed. Chapter 12 is devoted to the introduction of a queueing network tool, a stochastic Petri net tool, a tool based on Markov chains and a toolkit with many model types, and the facility for hierarchical modeling is also introduced. Each tool is described in some detail together with a simple example. Throughout the book we have provided many example applications of different algorithms introduced in the book. Finally, Chapter 13 is devoted to several large real-life applications of the modeling techniques presented in the book.

4

INTRODUCTION

Fig. 1.1 Flowchart mance problem.

describing

how to find the appropriate

chapter for a given perfor-

BASICS

1.2

BASICS OF PROBABILITY

OF PROBABILITY

AND

STATISTICS

5

AND STATISTICS

We begin by giving a brief overview of the more important definitions and results of probability theory. The reader can find additional details in books such as [AllegO, Fe1168,Koba78, Triv82]. We assumethat the reader is familiar with the basic properties and notations of probability theory. 1.2.1

Random Variables

A random variable is a function that reflects the result of a random experiment. For example, the result of the experiment “toss a single die” can be described by a random variable that can assumethe values one through six. The number of requests that arrive at an airline reservation system in one hour or the number of jobs that arrive at a computer system are also examples of a random variable. So is the time interval between the arrivals of two consecutive jobs at a computer system, or the throughput in such a system. The latter two examples can assumecontinuous values, whereas the first two only assumediscrete values. Therefore, we have to distinguish between continuous and discrete random variables. 1.2.1.1 Discrete Random Variables A random variable that can only assume discrete values is called a discrete random variable, where the discrete values are often non-negative integers. The random variable is described by the possible values that it can assumeand by the probabilities for each of these values. The set of these probabilities is called the probability mussfunction (pmf) of this random variable. Thus, if the possible values of a random variable X are the non-negative integers, then the pmf is given by the probabilities: pk = P(x

= k),

for k = 0, 1,2.. . ,

(1-l)

the probability that the random variable X assumesthe value Ic. The following is required: P(X = k) 2 0, c

P(X = k) = 1.

all k

For example, the following pmf results from the experiment “toss a single die” : P(X=k)=i,

forIc=1,2

,...,

6.

The following are other examples of discrete random variables: l

Bernoulli random variable: Consider a random experiment that has two possible outcomes, such as tossing a coin (Ic = 0,l). The pmf of the random

random variable X is given by: P(X l

= 0) = 1 -p

Binomial random is carried out n random variable The pmf of X is

and

with 0 < p < 1.

(1.2)

i

~“(1 -p)+lc,

0

k = O,l,...,

n.

w

The experiment with two possible outGeometric random variable: comes is carried out several times, where the random variable X now represents the number of trials it takes for the outcome 1 to occur (the current trial included). The pmf of X is given by: P(X

l

= 1) = p,

variable: The experiment with two possible outcomes times where successive trials are independent. The X is now the number of times the outcome 1 occurs. given by:

P(X= k)= l

P(X

= k) = p(l

Poisson random variable: pmf ) is given by:

- PI”-‘7

k = 1,2,. . . .

(1.4

The probability of having k events (Poisson

j’(X = k) = @&.e-,

k = 0, 1,2, . . . ; a > 0.

.

(1.5)

The Poisson and geometric random variables are very important to our topic; we will encounter them very often. Several important parameters can be derived from a pmf of a discrete random variable: l

Mean value or expected value: X=E[X]=~k?(X=k). all

(l-6)

k

The function of a random variable is another random variable with the expected value of:

all l

k

nth moments: X” = E[Xn] = c all

k”. P(X = k),

W)

k

that is, the nth moment is the expected value of the nth power of X. The first moment of X is simply the mean of X.

BASICS

l

OF PROBABILITY

AND

STATISTKS

7

nth central moment:

(X- xp = E[(X- E[X]y]= X(/k - X)“.P(X= k).

W)

all k

The nth central moment is the expected value of the nth power of the difference between X and its mean. The first central moment is equal to zero. l

The second central moment is called the variance of X: 0: = var(X)

= (X - 3Q2 = X2 -X2,

where crx is called the standard l

The coefficient of variation

(1.10)

deviation.

is the normalized cx=Y-.

standard

deviation:

*X

(1.11)

X

Information on the average deviation of a random variable from its expected value is provided by cx , ox, and var( X) . If cx = ox = var( X) = 0, then the random variable assumesa fixed value with probability one. Tab/e Random

Bernoulli Binomial Geometric Poisson

1.1 Propertiesof severaldiscreterandomvariables Variables

Parameter

P

w

52;

P

P&P)

w

w(l-P)

1 P

a

var(X)

1-P P 1-P

-

1-P

G

P2

a

cy

w

1-P

-1 Q

Table 1.1 gives a list of random variables, their mean values, variances, and the squared coefficients of variation for some important discrete random variables. A random variable X that can assume 121.2 Continuous Random Variables all values in the interval [a, b], where --00 5 a < b 5 +oo, is called a continuous random variable. It is described by its distribution function (also called CDF or cumulative distribution function): (1.12)

8

lNTRODlJCT/ON

which specifies the probability that the random variable X takes values less than or equal to x, for every x. Prom Eq. (1.12) we get for 2 < y: F,(x)

5 F,(Y)7

P(x < x 5 Y) = F,(Y)

- F, (4.

density function (pdf) f x (x ) can be used instead of the disThe probability tribution function, provided the latter is differentiable:

fx(x)

(1.13)

= y.

properties of the pdf are: fx (4 2 0 for all 2, 00 s -CO P(Xl

I

fx (4dx = 1,

x

L

x2)

=

x2 fx (x)dx, s Xl

~(x

= x) =

fx(x)dx

s

= 0,

2 cc

P(X

> x3) =

fx(x)dx.

s 23

Note that for so-called defective random variables [STP96] we have:

s

fx(x)dx

< 1.

-CC

The density function of a continuous random variable is analogous to the pmf of a discrete random variable. The formulae for the mean value and moments of continuous random variables can be derived from the formulae for discrete random variables by substituting the pmf by the pdf and the summation by an integral: l

Mean value or expected value: 03 x

= E[X]

=

s

x* fx (4dx

(1.14)

BASICS

OF PROBABILITY

AND

STATISTICS

9

and: (1.15)

l

nth moment: cc X”

= E[X”]

=

xn*

fx

(1.16)

(x)dx.

s --rx) l

nth central moment:

(X-X)” =E[(X -E[X]yy =px-T)“fx(x)dx. (1.17)

-CO

0 Variance: 0: = var(X) with ox as the standard 0 Coefficient

= (X - IT)’ = x2-

X2,

(1.18)

deviation.

of variation: cx

=- OX -.

(1.19)

X

A very well known and important continuous distribution function is the normal distribution. The CDF of a normally distributed random variable X is given by: F,(x)

= -$+J

] exp (-‘“LT2) --oo

du,

(1.20)

and the pdf by:

fx(2) = r&

exp (- (x2:?2)

.

X

The standard CDF:

normal distribution

is defined by setting x = 0 and ax = 1: @(x) = $= T. 1 exp (-z)

du,

--03

pdf:

c)(x) = -&.

II-

exp (-;)

.

(1.21)

10

INTRODUCTION

-3

-2

-1

0

1

2

3

X

Fig. 1.2

pdf of the standard

normal

random variable.

A plot of the preceding pdf is shown in Fig. 1.2. For an arbitrary normal distribution we have:

and

fx(x)

= 4

7

respectively. Other important continuous random variables are described as follows. (a) Exponential

Distribution

The exponential distribution is the most important and also the easiest to use distribution in queueing theory. Interarrival times and service times can often be represented exactly or approximately using the exponential distribution. The CDF of an exponentially distributed random variable X is given by Eq. (1.22):

o<x
with x =

X’ 1 2

if X represents interarrival times,

(1.22)

if X represents service times.

Here X or p denote the parameter of the random variable. In addition, for an exponentially distributed random variable with parameter X the following relations hold:

BASICS

OF PROBABILITY

pdf:

fx(z)

mean:

FE

variance:

var(X)

coefficient

of variation:

= Xe-

AND AX

STATISTICS

11

,

;, = $,

cx = 1.

Thus the exponential distribution is completely determined by its mean value. The importance of the exponential distribution is based on the fact that it is the only continuous distribution that possesses the memoryless property: P(XLU+tIX>u)=l-exp

(

-f

>

= P(X

5 t).

(1.23)

As an example for an application of Eq. (1.23), consider a bus stop with the following schedule: Buses arrive with exponentially distributed interarrival times and identical mean x. Now if you have already been waiting in vain for u units of time for the bus to come, the probability of a bus arrival within the next t units of time is the same as if you had just shown up at the bus stop, that is, you can forget about the past or about the time already spent waiting. Another important property of the exponential distribution is its relation to the discrete Poisson random variable. If the interarrival times are exponentially distributed and successive interarrival times are independent with identical mean F, then the random variable that represents the number of buses that arrive in a fixed interval of time [0, t) has a Poisson distribution with parameter a = t/s?. Two additional properties of the exponential distribution can be derived from the Poisson property: 1. If we merge n Poisson processeswith distributions for the interarrival times 1 - eSxtt, 1 -< i < - n, into one single process, then the result is a Poisson process for which the interarrival times have the distribution 1 - eext with X = Cy=, Xi (seeFig. 1.3).

Fig. 1.3 Merging of Poissonprocesses.

2. If a Poisson process with interarrival time distribution 1 - e- xt is split into n processesso that the probability that the arriving job is assigned

12

1NTRODUCTlON

to the ith process is qi, 1 5 i 5 n, then the ith subprocess has an interarrival time distribution of 1 - e-qiXt, i.e., n Poisson processeshave been created, as shown in Fig 1.4. ax / x -e

q2x . . .

Fig. 1.4 Splitting of a Poisson process.

The exponential distribution has many useful properties with analytic tractability, but is not always a good approximation to the observed distribution. Experiments have shown deviations. For example, the coefficient of variation of the service time of a processoris often greater than one, and for a peripheral device it is usually lessthan one. This observed behavior leads directly to the need to consider the following other distributions: (b) Hyperexponential

Distribution,

H,+

This distribution can be used to approximate empirical distributions with a coefficient of variation larger than one. Here k is the number of phases.

Fig. 1.5

A random variable with Hk distribution.

Figure 1.5 shows a model with hyperexponentially distributed time. The model is obtained by arranging k phaseswith exponentially distributed times and rates ~1,,22,. . . ,pk in parallel. The probability that the time span is given by the jth phase is qj , where C:=r qj = 1. However, only one phase can be occupied at any time. The resulting CDF is given by:

l+(x)

= &$(lj=l

e--I-Lq,

x 2 0.

(1.24)

BASICS

OF PROBABILITY

AND

STATISTICS

13

pdf:

mean:

variance:

coefficient of variation:

cx =

2&3-l i

j=l

2 1. ’

For example, the parameters PI, t32 of an Hz distribution can be estimated to approximate an unknown distribution with given mean x and sample coefficient of variation cx as follows: (1.25)

(1.26) The parameters qr and q2 can be assignedany values that satisfy the restrictions ql, q2 2 O,ql +qz = 1 and pi, ~2 > 0. (c) Erlang-k

Distribution,

Ek

Empirical distributions with a coefficient of variation less than one can be approximated using the Erlang-Ic distribution. Here Ic is the number of exponential phases in series. Figure 1.6 shows a model of a time duration that has an EI, distribution. It contains Ic identical phases connected in series, each with exponentially distributed time. The mean duration of each phase is x/i?, where x denotes the mean of the whole time span.

Fig. 1.6

A random variable with Ek distribution.

If the interarrival times of some arrival processlike our bus stops are identical exponentially distributed, it follows that the time between the first arrival and the (Ic + 1)th arrival is Erlang-k distributed.

14

INTRODUCTION

The CDF is given by: Fx(x)

x 2 0,k = 1,2....

= 1 -e

pdf:

fx(x)

=

kdkpx)k-l (k -

2 >

e-kpx,

I)!

0?k =

(1.27)

1 7 2 Y”‘,

1

xc

variance:

1, I-L var(X) = -+$,

coefficient of variation:

cx = 5

mean:

1

5 1.

If the sample mean x and the sample coefficient of variation cx are given, then the parameters k and ,IAof the corresponding Erlang distribution are estimated by: (1.28) and: 1 c$ Icy’

p=-

(d)

Hypoexponential

(1.29)

Distribution

The hypoexponential distribution ariseswhen the individual phasesof the Ek distribution are allowed to have assigned different rates. Thus, the Erlang distribution is a special case of the hypoexponential distribution. For a hypoexponential distributed random variable X with two phasesand the rates ~1 and ,u2 (~1 # ~2), we get the CDF as: l+(x)

= l-

P2

-e P2 -

-P1X +

Pl

Pl

P2

pdf:

fx(x)

mean:

XC

variance:

var(X)

coefficient of variation:

cx=h5Z
-

c-P2X

7

= s(emp2x

= -j!j + i,

fP2

0.

- evplx),

;+-$

Pl

x >

(1.30)

Pl

*

x > 0,

BASICS

OF PROBABILITY

AND

STATISTICS

15

The values of the parameters ~1 and ~2 of the hypoexponential CDF can be estimated given the sample mean x and sample coefficient of variation by:

with 0.5 5 c$ 5 1.

P2,

For a hypoexponential distribution pk, we get [Bega93]: *** >

with

k phases and the phase rates ~1,

k

pdf:

fx(x)

= ca&e;Aix,

2 > 0,

i=l k

with ai =

rI j=l,jzi

& pj - I&’

l 5 2 L Icy

mean:

coefficient

(e) Gamma

of variation:

cx =

Distribution

Another generalization of the Erlang-k distribution for arbitrary coefficient of variation is the gamma distribution. The distribution function is given by: X

cl+’ Fx(x)

=

(cxpu)“-l r(~)

f emapudu,

J

0, ct > 0,

(1.31)

0

with I’(a) =

x 2

u~-‘~~-~cZU,

a! > 0.

Jrn0

If a = k is a positive integer, then I’(k) = (k - l)! Thus the Erlang-k distribution can be considered as a special case of the gamma distribution:

16

INTRODUCTION

pdf: y=

variance:

1, P var(X) = --&,

coefficient of variation:

cx=-.

mean:

A

It can be seen that the parameters Q and ,Qof the gamma distribution can easily be estimated from cx and x: and

1

a=-.

CL (f)

Generalized

fig. 1.7

Erlang

Distribution

A random

variable with generalized

Erlang

distribution.

Rather complex distributions can be generated by combining hyperexponential and Erlang-Ic distributions; these are known as generalized Erlang distributions. An example is shown in Fig. 1.7 where 72parallel levels are depicted and each level j contains a series of rj phases connected, each with exponentially distributed time and rate rjpj. Each level j is selected with probability j. The pdf is given by: x 2 0.

(1.32)

Another type of generalized Erlang distribution can be obtained by lifting the restriction that all the phases within the same level have the same rate, or, alternatively, if there are non-zero exit probabilities assigned such

that remaining

phases can be skipped.

These generalizations

complicated equations that are not further described here.

lead to very

BASICS

(g) Cox Distribution,

cr,

(Branching

OF PROBABILITY

Erlang

AND

17

STATISTICS

Distribution)

In [Cox55] the principle of the combination of exponential distributions is generalized to such an extent that any distribution that possessesa rational Laplace transform can be represented by a sequence of exponential phases with possibly complex probabilities and complex rates. Figure 1.8 shows a model of a Cox distribution with k exponential phases.

Fig. 1.8 A randomvariable with Ck distribution.

The model consists of k phases in series with exponentially distributed times and rates ~1, ~2,. . . , ,~k. After phase j, another phase j + 1 follows with probability aj and with probability bj = 1 - aj the total time span is completed. As described in [SaCh81], there are two cases that must be distinguished when using the sample mean value x and the sample coefficient of variation cx to estimate the parameters of the Cox distribution: Case I: cx 5 1 To approximate a distribution with cx 5 1, we suggestusing a special Cox distribution with (see Fig. 1.9): /Lj

=/A

j=

l,...,k,

j=2,...,k-1.

CLj=l

From the probabilistic definitions, we obtain: x

mean:

=

h

+

k(l

-

bl)

I-L variance:

var(X)

squared coefficient of variation:

=

2 cx

=

7

k + bl(k - 1) (bl(1 - k) + k - 2) 7

P2 k + bl(k - 1) (bi(1 - k) + k - 2) [bl

+

k(l

-

h)12

’

In order to minimize the number of stages, we choose k such that: rli

k= k- . IG I

(1.33)

18

INTRODUCTION

Fig. 1.9

A random

variable with ck distribution

and cx 5 1

Then we obtain the parameters 131and ,LLfrom the preceding equations: b = 2kc$ + (ii - 2) - j/k2 + 4 - 41cc; 7 1 2(c$ + l)(k - 1) k - bl * (k - 1) P=

x

(1.34)

(1.35)

*

Case 2: cx > 1 For the case cx > 1, we use a Cox-2 model (see Fig. 1.lO)

Fig. 1.10

A random variable with Cox-2 distribution.

From classical formulae, we obtain: mean: variance:

var(x)

= l-4 + a/-w - a> ’ d 94

squared coefficient of variation:

2 _ Pi + ad@ - a> (~2 +wI)~ * cx -

We have two equations for the three parameters ~1, ~2, and a, and therefore obtain an infinite number of solutions. To obtain simple equations we choose: (1.36)

p1=$ and obtain from the preceding equations: 1 and a=ZC2. X

(1.37)

BASICS

(h) Weibull

OF PROBABILITY

AND

STATISTICS

19

Distribution

The Weibull distribution is at present the most widely used failure distribution. Its CDF is given by: Fx(x)

= 1 - exp(-(Xx)“),

pdf:

fx(x)

mean:

.x=;r

(1.38)

2 2 0.

= ~X(XX)“-~

exp(-(Xx)“),

X > 0,

1+: ) ( ) r(1 + 2/cu) cc = {I?(1 + l/o.)}2 - l*

squared coefficient of variation:

with the shape parameter a! > 0 and the scale parameter X > 0. (i) Lognormal

Distribution

The lognormal distribution is given by:

171(2)=m(l+y), pdf: mean:

f+)

= ---&

w+-@(x)

x>o. - J92/2a2>,

(1.39)

x > 0,

X = exp(X + a2/2),

squared coefficient of variation:

c$- = exp(a2) - 1,

where the shape parameter a! is positive and the scaleparameter X may assume any real value. The parameters a and X can easily be calculated from c$ and x: Q = -\/ln(cz + l), Q2 (1.40) 2 The importance of this distribution arises from the fact that the product of n mutually independent random variables has a log-normal distribution in the limit n -+ 00. In Table 1.2 the formulae for the expectation E[X], the variance war(X), and the coefficient of variation cx for some important distribution functions are summarized. Furthermore, in Table 1.3 formulae for estimating the parameters of these distributions are given. X=lnX--.

1.2.2

Multiple

Random

Variables

In some cases, the result of one random experiment determines the values of several random variables, where these values may also affect each other. The joint probability mass .function of the discrete random variables

20

INTRODUCTION

Table 1.2 Expectation important distributions.

E[X],

variance

var(X),

and coefficient

Distribution

Parameter

-WI

var(X)

Exponential

-1 P

1

CL cl,k k=l,Z,...

-1 P

kcL2

-cl! P

CL2

Erlang

Gamma

Hypoexponential

Pl

cx 1

CY

o<


L+L P2

2

4

k

Hyperexponential

k,I.Lz,qz

x:

k

2/A2cq1>1

= ;

i=l

i=l

&,X2,-.

cx of

/-L2 1

L+'

PlTP2

of variation

4

, X, is given by:

P(X1 = 21,x2 = x2,. . . ,xn = XJ

(1.41)

and represents the probability that Xi = xi, X2 = x2, . . . X, = x,. continuous case, the joint distribution function:

In the (1.42)

~x(x)=~(x1121,x2122,...,x7L<~~)

represents the probability that Xi 5 xi, X2 5 x2, . . . X, 5 x,, where X = X,) is the n-dimensional random variable and x = (xi, x2, . . . , x,). (X l,***, A simple example of an experiment with multiple discrete random variables is tossing a pair of dice. The following random variables might be determined: Xi

number that shows on the first die,

X2

number that shows on the second die, sum of the numbers of both dice.

X3=&+&,

independence The random variables Xi, X2, . . . , X, are called (statistically) independent if, for the continuous case:

1.2.2.1

P(XlLXl,X:! 5x2,...,Xn <xn) = P(X1 5 x1)* P(X2 < X2)‘. . . * P(X,

5 x,),

(1.43)

or the discrete case: P(X1=21,X2

=x2,...,Xn

=x,)

= P(X1 = Xl). P(X2 = X2)‘. , . * P(X,

= 2,).

(1.44)

BASICS

Table 1.3

Formulae

Distribution

for estimating

OF PROBABILITY

the parameters

Parameter

AND

of important

Calculation

STATISTICS

distributions.

of the Parameters

Exponential

Erlang

k

=

ceil(l/c$)

P

=

l/(+

cl,k

/c=1,2,...

kx)

Gamma

Hypoexponential

CL1= + [l+q-l

Hyperexponential

Wd

Pl,P2,Ql,Q2

= $ [l+&q-1

p2

41 +42

k

=

bl

=

=

1,/K!

> 0

ceil(l/c$) 2kc;+k-2-

cox

(cx

5 1)

2(C$

k, bz, puz bp p1

cox

(cx

> 1)

k,b,/a,m

7 + l)(k

+4-4kc, - 1)

=

b3 = . . . = bk-1

=

k - bl. (k - 1) /&’ = . . . = /.hk = x

k

=

2

b

=

c$[l-,/*I

=o,

k

bl, = 1

21

22

INTRODUCTION

Otherwise, they are (statisticuZZy) dependent. In the preceding example of tossing a pair of dice, the random variables Xr and X2 are independent, whereas Xr and Xa are dependent on each other. For example:

since there are 36 different possible results altogether probable. Because the following is also valid:

that are all equally

=-t6,

P(x,=i)*P(x2=j)+;

P(Xi = i, X2 = j) = P(Xi = i)- P(X2 = j) is true, and therefore Xr and X2 are independent. On the other hand, if we observe the dependent variables Xi and Xs, then:

since Xr and X2 are independent. P(X1 = 2) = ;

Having:

and

P(X3=4)=P(X1=1,Xz=3)+P(X1=2,X2=2) +P(X,=3,Xz=1)=$& results in P(Xr = 2). P(Xa = 4) = 6. & = $, which shows the dependency of the random variables Xr and X3 : P(X1 = 2,X3 = 4) # P(X1 = 2)+(X3 1.2.2.2

Conditional

A conditional probability:

Probability P(X1

= 4).

=x1

1 x2 =22,.*.,X,=x,)

is the probability for Xr = 21 under the conditions X2 = x2, X3 = x3, etc. Then we get: P(X1

= Xl =

1 x2 =

x2,.

, . ) x,

= xn)

P(X1=21,X2 =x2 )...) x, =x,) P(X2 =x2,...,xn

(1.45)

=x,) *

For continuous random variables we have: P(X1

I =

P(Xl

Xl

1 x2

L

1X1,X2 P(X2

x2,.

. . ) x,

5

x?J

Lx2,...Jn L

x2,-..&I

(1.46)

LGL) I

xn)

*

BASICS

OF PROBABILITY

AND

STATISTICS

23

We demonstrate this also with the preceding example of tossing a pair of dice. The probability that Xa = j under the condition that Xi = i is given by:

= j,x, = i) P(X3 = j ) x1 = i) = P(X3 P(X1= i) * For example, with j = 4 and i = 2: l/36 1 P(X3 = 4 ) x1 = 2) = = l/6 6’ If we now observe both random variables Xi and X2, we will see that because of the independence of Xi and X2,

P(X1 =j)

x2 =i)

=

P(X1=

j, x2 = i) P(X2 = i) ’

= P(X1=

j)*P(X2 P(X2 = i)

= P(X1=

= i) ’

j).

1.2.2.3 important Relations The following relations concerning multiple random variables will be used frequently in the text: l

The expected value of a sum of random variables is equal to the sum of the expected values of these random variables. If cl, . . . , c, are arbitrary constants and Xi, X2, . . . , X, are (not necessarily independent) random variables, then:

rn

1

n

E c cixi = c CiE[XJ. i i=l 1 i=l l

(1.47)

If the random variables Xi, X2, . . . , X, are stochastically independent, then the expected value of the product of the random variables is equal to the product of the expected values of the random variables:

rn

ln

(1.48)

l

The covariance of two random variables X and Y is a way of measuring the dependency between X and Y. It is defined by: cov[X, Y] = E [(X - E[X])(Y = E[X.Y]

- E[Y])] = (X - r?)(Y - L) -- E[X]. E[Y] = XY - X-Y. (1.49)

24

INTRODUCTION

If X = Y, then the covariance is equal to the variance: cov[X, X] = x2 - X2 = var(X) If X and Y are statistically

independent,

= a$.

then Eq. (1.48) gives us:

E[X* Y] = E[X]LqY], and Eq. (1.49):

cov(X, Y) = 0. In this case, X and Y are said to be uncorrelated. l

The correlation coeficient of standard deviations:

is the covariance

cor[X, Y] = Therefore,

to the product

cov[X, Y] Ux’Uy -

(1.50)

if X = Y, then: cor[X,X]

l

normalized

= 2

= 1.

The variance of a sum of random variables can be expressed covariance:

using the

n

var

[c

i=l

ciXi

1= 2

For two independent

cT var(Xi)

+ 2.2

i=l

i=l

(uncorrelated)

[c

i=l

ciXi

cicj COV[Xi, Xj].

(1.51)

random variables we get:

n

var

f: j=i+1

= 2

1

cf var(Xi).

(1.52)

i=l

1.2.2.4 The Central Limit Theorem Let Xr , X2, . . . , X, be independent, identically distributed random variables with an expected value of E[X,] = z, and a variance of var( Xi) = a$. Then their arithmetic mean is defined by:

sn=;gxi. i=l We have E[Sn] = x and var(&) = OS/n. Regardless of the distribution of Xi, the random variable S, approaches, with increasing n, a normal distribution with a mean of x and a variance of var(X) /n: (1.53)

BASKS

1.2.3

OF PROBABILITY

AND

25

STATISTICS

Transforms

Determining the mean values, variance, or moments of random variables is often rather tedious. However, these difficulties can sometimes be avoided by using transforms, since the pmf of a discrete or the density function of a continuous random variable are uniquely determined by their x- or Laplace transform, respectively. Thus if two random variables have the same transform, they also have the same distribution function and vice versa. In many cases, the parameters of a random variable are, therefore, more easily computed from the transform. 1.2.3.1 x-Transform Let X be a discrete random variable with the probability massfunction pk = P(X = k) for k = 0, 1,2,. . . , then the x-transform of X is defined by: G,(z)

= Fp/+z”

with jz[ 5 1.

(1.54)

k=O

The function G,(z) is also called the probability generating function. Table 1.4 specifies the probability generating functions of some important discrete random variables. If the x-transform G, (x) of a random variable is given, then the probabilities pk can be calculated immediately: (1.55)

PO = G,(O),

1 d”G,(z) pk=

jJ’

k = 1,2,3,. . * *

7

&k

(1.56)

z=o

And for the moments: (1.57) (1.58) x"=

k-l

d”G, (4 dx’”

z=l

ak,,i’ 3,

+x

k=3,4,...,

(1.59)

i=l

with

3ak,l = (-l)“(k ak,k-l ak,i

- I)!,

k = 3,4,. . . ,

k(k - 1) 2 ’

=

~

=

ak-l,i-1

-

k = 3,4,. . . ) (k

-

l)ak-l,i,

k = 4,5, . . . and i = 2, . . . , k - 2.

26

INTRODUCTION

Table 1.4 Generating functions of some discrete random variables Random

Variable

Parameter

Binomial

G,

n,p

Geometric

P

Poisson

a

The generating function product of the generating CF=, Xi we get:

u

(4

- P> +

P4”

PZ

1 - (1 -

p)z

pb-1)

of a sum of independent random variables is the functions of each random variable, so for X =

G,(z) = fi G&).

(1.60)

i=l

1.2.3.2 Lap/ace Transform The Laplace transform plays a similar role for non-negative continuous random variables as the z-transform does for discrete random variables. If X is a continuous random variable with the density function fx(z), then the Lap&e transform (LT) of the density function (pdf) of X is defined by:

L,(s)

= mfx(x)e-s5dz, I

1emsI< _ 1,

(1.61)

0

where s is a complex parameter. Lx(s) is also called the Laplace-Stieltjes transform (LST) of the distribution function (CDF) and can also be written 03 Lx(s) =

I

e-““dFx(z).

(1.62)

0

LST of a random variable X will also be denoted by X” (s). The moments can again be determined by differentiation: ,

k=1,2

,....

(1.63)

s=o The Laplace transforms of several pdfs of well-known continuous random variables as well as their moments are given in Table 1.5.

BASICS

Table 1.5 Random

Laplace transform Variable

OF PROBABILITY

AND

and moments of several continuous

Parameter

27

STATISTICS

random variables X”

L&)=x-(s)

Exponential k(k

Erlang

k,X

+ 1). . . (k + n - 1) An

CY(LY+ 1). . . (CL:+ n - 1) An

Gamma

Hyperexponential

x

Hypoexponential

Analogous to the x-transform, the Laplace transform of a sum of independent random variables is the product of the Laplace transform of these random variables: (1.64) i=l

More properties and details of Laplace and z-transforms are listed in [Klei75]. 1.2.4

Parameter

Estimation

Probability calculations discussedin this book require that the parameters of all the relevant distributions are known in advance. For practical use, these parameters have to be estimated from measured data. In this subsection, we briefly discussstatistical methods of parameter estimation. For further details see [Triv82]. Definition 1.1 Let the set of random variables Xi, X2, . . . , X, be given. This set is said to constitute a random sample of size n from a population with associated distribution function Fx(x). It is presupposed that the random variables are mutually independent and identically distributed with distribution function (CDF) Fxi (x) = Fx(x) V’i, x. We are interested in estimating the value of some parameter 6’ (e. g. , the mean or the variance) of the population based on the random samples. Definition 1.2 A function 6 = 6(X1, X2, . . . , X,) is called an estimator of the parameter 8, and an observed value 6 = 6(x,, 22 . . . , x~) is known as

28

INTRODUCTION

an estimate of 8. An estimator

6 is considered

unbiased, if:

E[QXl,

x2, * * . )X,)] = 8. It can be shown that the sample mean X defined as:

,2exi

(1.65) i=l

is an unbiased estimator S2 defined as:

of the population

mean p, and the sample variance

s2 = 1 &Xi n - l i=l is an unbiased estimator

- 7q2

of the population

(1.66)

variance 02,

Definition 1.3 An estimator 6 of a parameter 8 is said to be consistent if 6 converges in probability to 8 (so-called stochastic convergence); that is: lim P(lC3 - 81 2 e) = OV/E> 0.

n+oo

1.2.4.1

Method

of Moments

The Icth sample moment of a random variable

X is defined as: n x’” ML =cx--$

k= 1,2 ,....

(1.67)

i=l

To compute one or more parameters of the distribution of X, using the method of moments, we equate the first few population moments with their corresponding sample moments to obtain as many equations as the number of unknown parameters. These equations can then be solved to obtain the required estimates. The method yields estimators that are consistent, but they may be biased. As an example, suppose the time to failure of a computer system has a gamma distributed pdf: pz”-le-x~ fx(4

,

=

k= 1,2 ,...,

(1.68)

wd

with parameters X and a. We know that the first two population moments are given by: and I_L~= 5 + ~1~. Pl = ; Then the estimates i and & can be obtained by solving:

Hence: &!=

Ml2 M2 - M12’

and i =

Ml

M2 - M12’

(1.69)

BASICS OF PROBABILITY

AND STATISTICS

29

The principle of this method is to 1.2.4.2 Maximum- Likelihood Estimation select as an estimate of 6’the value for which the observed sample is most likely to occur. We define the likelihood function as the product of the marginal densities:

,13,) is the vector of parameters to be estimated. Under where t9 = (&,&,... some regularity conditions, the maximum-likelihood estimate of 8 is the solution of the simultaneous equations: ALP) -

dei

=o,

i=

1,2 ,...) Ic.

As an example, assume the interarrival time, X, of a system is exponentially distributed with arrival rate X. To estimate the arrival rate X from a random sample of n interarrival times, we define: L(X) = fiAexp(-Xxi,) i=l

= X’“exp(-AF

zi),

(1.70)

i=l

dL = nAna dX

from which we get the maximum-likelihood estimator of the arrival rate that is equal to the reciprocal of the sample mean x. 1.2.4.3 Confidence Intervals So far we have concentrated on obtaining the point estimates of a desired parameter. However, it is very rare in practice for the point estimate to actually coincide with the exact value of the parameter 8 under consideration. In this section, we instead consider the evaluation of an interval estimate. We construct an interval, called the confidence interval, in such a way that we have a certain confidence that this interval contains the true value of the unknown parameter 8. In other words, if the estimator 6 satisfies the condition:

P@-~~<~&+E~)=Y,

(1.72)

then the interval A(8) = (6 - cl,6 + ~2) is a 100 x y percent confidence interval for the parameter 8. Here y is called the confidence coefficient (the probability that the confidence interval contains S). Consider obtaining confidence intervals when using the sample mean x as the estimator for the population mean. If the population distribution is normal with unknown mean ,Q and a known variance g2, then we can show that the sample mean is normally distributed with mean ,Qand variance 02/n, so that the random variable 2 = (57 - ,LL)/ (a/+) has a standard normal distribution N(0, 1). To determine a 100 x y percent confidence interval for

30

/NTROoUCnON

the population mean ,LL,then we find a number z,12 (using N(0, 1) tables) such that P(Z > z,,s) = a/2, where a = 1 - y. Then we get: < 2 <

PC-%/2

&/2)

(1.73)

=Y*

We then obtain the 100 x y percent confidence interval as: < (X-P)I(+m

-G/2

<

(1.74)

G/2

or: (1.75) The assumption that the population is normally distributed does not always hold. But when the sample size is large, by central limit theorem, the statistic CT- PL)l(~ldT n is asymptotically normal (under appropriate conditions). Also, Eq. (1.75) requires the knowledge of the population variance u2. When a2 is unknown, we use the sample variance S2 to get the approximate confidence interval for p: (1.76) When the sample size is relatively small and the population is normal distributed, then the random variable T = (x - p)/(S/fi) has the student t distribution with n - 1 degrees of freedom. We can then obtain the lOO( 1 - o) percent confidence interval of /J from: x

- L1;,/2-

where P(T > tn-pa,2) 1.2.5

Order

$

< p < x

+ t,-pap-

g

(1.77)

= a/2.

Statistics

Let X1, X2 . . . , X, be mutually independent, identically distributed continuous random variables, each having the distribution function F(.) and density f(.). Let Yr, Ys,. . . , Y, be random variables obtained by permuting the set ,X, so as to be in increasing order. Thus, for instance: x1,x2,... Yr = min{Xr,

X2,. . . ,X,}

and: Y, = max{Xr,X2,.

. . ,Xn}.

The random variable Yk is generally called the kth -order statistic. Because X, are continuous random variables, it follows that Yr < YZ < x1,x2,..., . . . < Y, (as opposed to Yr 5 Y2 < . . . < Yn) with a probability of one.

BASICS

OF PROBABILITY

AND

STATISTICS

31

As examples of the use of order statistics, let Xi be the time to failure of the ith component in a system of n independent components. If the system is a series system, then Yr will be the system lifetime. Similarly, Y, will denote the lifetime of a parallel system and Y,-,+I will be the lifetime of an m-out-of-n system. The distribution function of Yk is given by:

FYdY)

= 2

&%)[l

-oo
- F(y)]“-?

(1.78)

j=k

In particular, the distribution functions of Y, and Yr can be obtained from Eq. (1.78) as:

--oo< Y< 00,

= F%m

Fy,(Y)

and F&(Y) = 1 - [I - F(y)]“, 1.2.6

Distribution

-oo < y < 00.

of Sums

Assume that X and Y are continuous random variables with joint probability density function f. Suppose we are interested in the density of a random variable 2 = @(X, Y). The distribution of 2 may be written as: F&)

= P(Z 5 x) =

fh JJ A*

Y)dXdY,

(1.79)

where A, is a subset of R2 given by:

Az =

{(x,Y>

I Q(x, Y> L 21

= @-“((-co,

x]).

A function of special interest is 2 = X + Y with

AZ = {(x, Y)

I x + Y 5 4,

which is the half-plane to the lower left of the line x+y = x, shown in Fig. 1.11. Then we get:

JJf(x, = f(x, JJ -m -ccl

Fz(z) =

Y)dXdY

00

z-x

YPYdX*

32

INTRODUCTION

Fig. 1.11

Area of integration

for the convolution

of X and Y.

Making a variable substitution y = t - LC,we get: co rz(~)

z

= JJ -co .z

-

x)dtdx

f(x,t

-

x) dxdt

00

= JJ -cm

=

f(x,t -cm

-co

Jfz(Wt.

Thus the density of 2 is given by:

fz(~> = 7 f(x, x - x)dx, -CO

-co<x<m.

(1.80)

Now if X and Y are independent random variables, then f (x, y) = fx (x) fu (y), and Formula (1.80) reduces to:

f.h>

= 7 fx(x)fy(x --00

- x)dx,

-oo<X<~.

(1.81)

Further, if both X and Y are nonnegative random variables, then:

f&d

= /- fx(x)fu(z 0

- x)dx,

o
(1.82)

BASICS

OF PROBABILITY

AND

STATISTICS

33

This integral is known as the convolution of fx and fy . Thus, the pdf of the sum of two non-negative independent random variables is the convolution of their individual pdfs. If xr, x2,..., X, are mutually independent, identically distributed exponential random variables with parameter X, then the random variable Xi + x2 + * * * + X, has an Erlang-r distribution with parameter X. Consider an asynchronous transfer mode (ATM) switch with cell interarrival times that are independent from each other and exponentially distributed with parameter X. Let Xi be the random variable denoting the time between the (i - 1)st and ith arrival. Then 2, = Xi +X2 +. . . +X, is the time until the rth arrival and has an Erlang-r distribution. Another way to obtain this result is to consider N,, the number of arrivals in the interval (0, t]. As pointed out earlier, Nt has a Poisson distribution with parameter At. Since the events [Z, > t] and [N, < ~1 are equivalent, we have: P(&

> t) = P(N, < r)

which implies that: F&(t)

= P(.G = l-c,,e

I t) ?--I @t)j

-At >

j=o

*

which is the Erlang-r distribution function. If X N EXP(Xr), Y N EXP(&), X and Y are independent, and Xi # X2, then 2 = X + Y has a two-stage hypoexponential distribution with two phasesand parameters Xi and X2. To seethis: 2> 0

(by Eq. (1.82))

= Jjle-AlXX2e-X2(z-z)dZ .I 0

WQ + -ewXIZ

h&i = -e-‘2’ x1-

x2

x2 -x1

(see Eq. (1.30)).

34

INTRODUCTION

Fig. 1.12

Hypoexponential

A more general

version

distribution

as a series of exponential

stages.

of this result follows: Let 2 = 2 Xi, where i=l

. , X, are mutually independent and Xi is exponentially distributed with parameter Xi (Xi # Xj for i # j). Then 2 has a hypoexponential distribution with r phasesand the density function: x1,x2,..

(1.83) where: r ai=

rI j.l

- XLi Xj - Ai ’

l
(1.84)

j#i Such a phase type distribution can be visualized as shown in Fig. 1.12 (see also Section 1.2.1.2).

2 Markov Chains 2.1

MARKOV

PROCESSES

Markov processes provide very flexible, powerful, and efficient means for the description and analysis of dynamic (computer) system properties. Performance and dependability measures can be easily derived. Moreover, Markov processes constitute the fundamental theory underlying the concept of queueing systems. In fact, the notation of queueing systems has been viewed sometimes as a high-level specification technique for (a sub-class of) Markov processes. Each queueing system can, in principle, be mapped onto an instance of a Markov process and then mathematically evaluated in terms of this process. But besides highlighting the computational relation between Markov processes and queueing systems, it is worthwhile pointing out also that fundamental properties of queueing systems are commonly proved in terms of the underlying Markov processes. This type of use of Markov processes is also possible even when queueing systems exhibit properties such as non-exponential distributions that cannot be represented directly by discrete-state Markov models. Markovizing methods, such as embedding techniques or supplementary variables, can be used in such cases. Here Markov processes serve as a mere theoretical framework to prove the correctness of computational methods applied directly to the analysis of queueing systems. For the sake of efficiency, an explicit creation of the Markov process is preferably avoided. 2.1.1

Stochastic

and Markov

Processes

There exist many textbooks like those from King [KingSO], Trivedi [Triv82], Allen [AllegO], G ross and Harris [GrHa85], Cinlar [Cin1?5], Feller (his two

volume classic [FeUSS]), and Howard [Howa71] that provide excellent intro35

36

MARKOV CHAINS

ductions into the basics of stochastic and Markov processes. Besides the theoretical background, many motivating examples are also given in those books. Consequently, we limit discussion here to the essentials of Markov processes and refer to the literature for further details. Markov processes constitute a special, perhaps the most important, subclass of stochastic processes, while the latter can be considered as a generalization of the concept of random variables. In particular, a stochastic process provides a relation between the elements of a possibly infinite family of random variables. A series of random experiments can thus be taken into consideration and analyzed as a whole. Definition 2.1 A stochastic process is defined as a family of random variables {Xt : t E T} where each random variable Xt is indexed by parameter t E 57, which is usually called the time parameter if T G IK+ = [0, co). The set of all possible values of Xt (for each t E T) is known as the state space S of the stochastic process. If a countable, discrete-parameter set T is encountered, the stochastic process is called a discrete-parameter process and 7’ is commonly represented by (a subset of) No = {O,l,. . .}; otherwise we call it a continuous-parameter process. The state space of the stochastic process may also be continuous or discrete. Generally, we restrict ourselves here to the investigation of discrete state spaces and in that case refer to the stochastic processes as chains, but both continuous- and discrete-parameter processes are considered. Continuous-parameter stochastic processes can be probaDefinition 2.2 bilistically characterized by the joint (cumulative) distribution function (CDF) Fx(s; t) for a given set of random variables {Xt, , Xt, , . . . , Xt,}, parameter vector t = (tl, t2, . . . , tn) E Iw”, and state vector s = (si, ~2,. . . , sn) E Iw”, where tl < t2 < . . . < t,: Fx(s;t)

=P(Xt,

<

Sl,&

I

S2,-.Jt,

I

sn>.

(24

The joint probability density function, pdf:

PFx(s; t) fx(s; t) = as&~ . . . ds, is defined correspondingly if the partial derivatives exist. If the so-called Markov property is imposed on the conditional CDF of a stochastic process, a Markov process results: A stochastic process {Xt : t E T} constitutes a Murkov Definition 2.3 process if for all 0 = to < tl < . . . < t, < tn+l and all si E S the conditional CDF of Xt,+l depends only on the last previous value Xt, and not on the earlier values Xt, , X,, , . . . , Xtnel : P(Xt =

n+l L &+1

p(&,+l

I

I Xtn

= %,xt,-l

%+1

I Xtn

= 4.

= h-1,.

* * A,

= so) cw

MARKOV PROCESSES

37

This most general definition of a Markov process can be adopted to special cases. In particular, we focus here on discrete state spaces and on both discrete- and continuous-parameter Markov processes. As a result, we deal primarily with continuous-time Markov chains (CTMC), and with discretewhich are introduced in the next section. time Markov chains (DTMC), Finally, it is often sufficient to consider only systems with a time independent, i.e., time-homogeneous, pattern of dynamic behavior. Note that timehomogeneous system dynamics is to be discriminated from stationary system behavior, which relates to time independence in a different sense. The former refers to the stationarity of the conditional CDF while the latter refers to the stationarity of the CDP itself, Definition 2.4 Letting to = 0 without loss of generality, a Markov process is said to be time-homogeneous if the conditional CDF of Xt,+l does not depend on the observation time, that is, it is invariant with respect to time epoch t,: P(&n+l

2.1.2

Markov

I

%+1

I xt,

= sn) = qG,+&

I

h&+1 I x0

= Sn).

(2.3)

Chains

Equation (2.2) d escribes the well-known Markov property. Informally this can be interpreted in the sense that the whole history of a Markov chain is summarized in the current state Xtn. Equivalently, given the present, the future is conditionally independent of the past. Note that the Markov property does not prevent the conditional distribution from being dependent on the time variable t,. Such a dependence is prevented by the definition of homogeneity (see Eq. (2.3)). A um‘q ue characteristic is implied, namely, the sojourn time distribution in any state of a homogeneous Markov chain exhibits the memoryless property. An immediate, and somewhat curious, consequence is that the mean sojourn time equals the mean residual and the mean elapsed time in any state and at any time [Triv82]. If not explicitly stated otherwise, we consider Markov processes with discrete state spaces only, that is, Markov chains, in what follows. Note that in this case we are inclined to talk about probability mass functions, pmf, rather than probability density functions, pdf. Refer back to Sections 1.2.1.1 and 1.2.1.2 for details. 2.1.2.1 Discrete-Time Markov Chains We are now ready to proceed to the formal definition of Markov chains. Discrete-parameter Markov chains are considered first, that is, Markov processes restricted to a discrete, finite, or countably infinite state space, S, and a discrete-parameter space T. For the sake of convenience, we set T 2 lVe. The conditional pmf reflecting the Markov property for discrete-time Markov chains, corresponding to Eq. (2.2), is summarized in the following definition:

MARKOV CHAINS

A given stochastic process (X0, Xi,. . . , Xn+i, . . .} at the Definition 2.5 consecutive points of observation 0, 1, . . . , n + 1 constitutes a DTMC if the following relation 011the conditional pmf, that is, the Markov property, holds for all n E Ne and all .si E S:

P(X,+1 = s,+1 1x, = s,, X,-l

= s,-1, . . . ,x0 = so) (2.4

=

q&a+1

=

ha+1

I xx

=

sn).

Given an initial state se, the DTMC evolves over time, that is, step by step, according to one-step transition probabilities. The right-hand side of Eq. (2.4) reveals the conditional pmf of transitions from state s, at time step n to state s,+i at time step (n + 1). Without loss of generality, let an d wri t e conveniently the following shorthand not ation for S = {0,1,2,...} the conditional pmf of the process’s one-step transition from state i to state j at time n: p,‘:‘(n) = P(X,+l

= s,+l = j 1X, = sn = i).

P-5)

In the homogeneous case, when the conditional pmf is independent of epoch n, Eq. (2.5) reduces to: p!!) = p!!)(n) = P(X n+l = j 1X, = i) = P(X1 = j 1X0 = i), 2.7 a.7

Vn E T. (2.6)

For the sake of convenience, we usually drop the superscript, so that pij = ~13) refers to a one-step transition probability of a homogeneous DTMC. Starting with state i, the DTMC will go to some state j (including the possibility of j = i), so that it follows that Cj pij = 1, where 0 5 pij 5 1. The one-step transition probabilities pij are usually summarized in a nonnegative, stochastic’ transition matrix P:

p

= p(l)

=

bij]

=

PO0

PO1

po2

**’

;;;

;;;

g:;

: 1:

I

.

. . .

. . .

.

.

.I

Graphically, a finite-state DTMC is represented by a state transition diagram, a finite directed graph, where state i of the chain is depicted by a vertex, and a one-step transition from state i to state j by an edge marked with one-step transition probability pij. As an example, consider the one-step transition probability matrix in Eq. (2.7) with state space S = (0, 1) and the corresponding graphical representation in Fig. 2.1.

‘The

elements

in each row of the matrix

sum

up to

1.

MARKOV PROCESSES

Example 2.1 The one-step transition probability DTMC in Fig. 2.1 is given by:

39

matrix of the two-state

Conditioned on the current DTMC state, a transition is made from state 0 to state 1 with probability 0.25, and with probability0.75, the DTMC remains in state 0 at the next time step. Correspondingly, a transition occurs from state 1 to state 0 with probability 0.5, and with probability 0.5 the chain remains in state 1 at the next time step.

Fig. 2.1

Example of a discrete-time Markov chain referring to Eq. (2.7).

Repeatedly applying one-step transitions generalizes immediately to n-step transition probabilities. More precisely, let p$‘(Ic, Z) denote the probability that the Markov chain transits from state i at time Ic to state j at time Z in exactly n = 1 - Ic steps:

Again, the theorem of total probability applies for any given state i and any given time values k and Z such that Cj piy)(b, 1) = 1, where 0 < piy)(k, 1) 5 1. This fact, together with the Markov property, immediately leads us to a procedure for computing the n-step transition probabilities recursively from the one-step transition probabilities: The transition of the process from state i at time Ic to state j at time Zcan be split into subtransitions from state i at time k to an intermediate state2 h, say, at time m and from there, independently of the history that led to that state, from state h at time m to state j at time I, where k < m < Z and n = Z - k. This condition leads to the well known system of Chapman-Kolmogorov equations: p$)(k, Z) = c

p,(;-“) (k,m)pL3ym)(m 7Z)7 0 5 k < m < 1.

cw

hES

Note that the conditional independence assumption, i.e., the Markov property, is reflected by the product of terms on the right-hand side of Eq. (2.9).

2The Markov chain must simply traverse some state at any time.

40

MARKOV CHAINS

Similar to the one-step case, the n-step transition probabilities can be simplified for homogeneous DTMC such that pi:) = p$) (Ic, Z) depend only on the difference n = I - k and not on the actual values of k and I: p!,“’ = P(x~+,

= j 1XI, = i) = P(X,

= j 1X0 = i),

Under this condition, the Chapman-Kolmogorov DTMC simplifies to: p!?) %.I

=

C

(2.10)

Eq. (2.9) for homogeneous

0 <

p,(;l”)p~-m’,

‘dk E T.

m

<

n.

&S

Because Eq. (2.11) holds for all m < n, let m = 1 and get

(n)-_c Pi, (1) (n-1) Pij P&j * hES

With Pen) as the matrix of n-step transition probabilities ply’, Eq. (2.11) can be rewritten in matrix form for the particular case of m = 1 as Pen) = p(l)p(n-1) = pp(n-1). A pp 1ying this procedure recursively results in the following equation3: ~(4 = pp(n--1) = pn.

(2.12)

The n-step transition probability matrix can be computed by the (n - l)-fold multiplication of the one-step transition matrix by itself. Referring back to the example in Fig. 2.1 and Eq. (2.7), the Example 2.2 four-step transition probability matrix, for instance, can be derived according to Eq. (2.12): p(4) = pp(“) = (‘b;;

=

= p2pw og2Prz)

0.65625 0.67188

= (y-$7;

0.34375 0.32813

y-y;;)

0.66406 0.66797

PPW

(2*13)

0.33594 0.33203 > *

Ultimately, we wish to compute the pmf of the random variable X,, that is, the probabilities v;(n) = P(X, = i) that the DTMC is in state i, at time step n. These probabilities, called transient state probabilities at time n, will then allow us to derive the desired performance measures. Given the n-step transition probability matrix P cn), the vector of the state probabilities at time

31t is important

to keep in mind

that

P cn)

= Pn holds

only

for homogeneous

DTMC.

MARKOV PROCESSES

n, y(n) =

@0(n),

vi(n),

v2(n),

, . .>

the initial probability vector

can be obtained by unconditioning

y(O)

=

(vo(O),

e(O),

vz(O),

41

Pen) on

- * .>:

y(n) = v(O)P (4 = v(o)Pn = v(?2 - l)P.

(2.14)

Note that both v(n) and v(0) are represented as row vectors in Eq. (2.14). Assume that the DTMC from Eq. (2.7) under investigation Example 2.3 is initiated in state 1, then the initial probability vector v(l)(O)=(O, 1) is to be applied in the unconditioning according to Eq. (2.14). With the already computed four-step transition probabilities in Eq. (2.13) the corresponding pmf y(l) (4) can be derived: “(l)(4) Example

2.4

0.66797 0.66406

= (0,l)

Alternatively,

0.33203 = (0.66406,0.33594). 0.33594 >

with another initial probability

vector:

“(2)(O) = ($, ;> = (0.66,0.33) , according to Eq. (2.14), we would get: ~(~~(4) = (0.66,0.33) as the probabilities at time step 4. Of particular importance are homogeneous DTMC on which a so-called stationary probability vector can be imposed in a suitable way: State probabilities u = (~0, ~1,. . . vi,. . .) of a discrete-time Definition 2.6 Markov chain are said to be stationary, if any transitions of the underlying DTMC according to the given one-step transition probabilities P = bij] have no effect on these state probabilities, that is, vj = ‘& uipij holds for all states j E S. This relation can also be expressed in matrix form: u = VP,

c

vi = 1.

(2.15)

iES

Note that according to the preceding definition, more than one stationary pmf can exist for a given, unrestricted, DTMC. By substituting the one-step transition matrix from Eq. (2.7) Example 2.5 in Eq. (2.15), it can easily be checked that: uc2) = (;, 3) = (0.66,0.33) , is a stationary probability

vector while Y(‘) = (0,l) is not.

42

MARKOV CHAINS

For an efficient analysis, we are interested in the limiting Definition 2.7 state probabilities 6 as a particular kind of stationary state probabilities, which are defined by: fi = &nrV(n)

= $r& v(0)P(nJ = y(O) lim Pen) = y(O)p. n-+03

(2.16)

As n + cc, we may require both the n-step transition Definition 2.8 probability matrix Pen) and the state probability vector v(n) to converge independently of the initial probability vector v(0) to %’ and V, respectively. Also, we may only be interested in the case where the state probabilities Vi > 0, Vi E S, are strictly positive and xi Yi = 1, that is, 5 constitutes a pmf. If all these restrictions apply to a given probability vector, it is said to be the unique steady-state probability vector of the DTMC. Example 2.6 tion probabilities

Returning to Example 2.1, we note that the n-step transiconverge as n -+ 00:

With this result, the limiting state probability vector G, Example 2.7 which is independent of any initial probability vector Y(O), can be derived according to Eq. (2.16):

2, = (0.66,0.33). Example 2.8 Since all probabilities in the vector (0.66,0.33) are strictly positive, this vector constitutes the unique steady-state probability vector of the DTMC. Eventually, the limiting state probabilities become independent of time steps, such that once the limiting probability vector is reached, further transitions of the DTMC do not change this vector, i.e., it is stationary. Note that such a probability vector does not necessarily exist for all DTMCs. If Eq. (2.16) holds and fi is independent of v (0)) it follows that the limit p(n) = [p!“‘] is independent of time n and of index i. All rows of 6 would be identiczl, that is, the rows would match element by element. Furthermore, the jt h element j&j of row i equals zP~for all i E S:

(2.17)

If the unique steady-state probability vector of a DTMC exists, it can be determined by the solution of the system of linear Eqs. (2.15), so that I? need

MARKOV PROCESSES

43

not be determined explicitly. From Eq. (2.14), we have y(n) = v(n - l)P. If the limit exists, we can take it on both sides of the equation and get: lim v(n) = ZI = lim v(n - l)P = C/p.

n+cc

(2.18)

n+cc

In the steady-state case no ambiguity can arise, so that, for the sake of convenience, we may drop the annotation and refer to steady-state probability vector by using the notation u instead of t. Steady-state and stationarity coincide in that case, i.e., there is only a unique stationary probability vector. The computation of the steady-state probability vector u of a DTMC is usually significantly simpler and less expensive than a time-dependent computation of v(n). It is therefore the steady-state probability vector of a DTMC that is preferably taken advantage of in modeling endeavors. But a steadystate probability vector does not exist for all DTMCS.~ Additionally, it is not always appropriate to restrict the analysis to the steady-state case, even if it does exist. Under some circumstances time-dependent, i.e., transient, analysis would result in more meaningful information with respect to an application. Transient analysis has special relevance if short-term behavior is of more importance than long-term behavior. In modeling terms, “short term” means that the influence of the initial state probability vector Y(O) on v(n) has not yet disappeared by time step n. Before continuing, some simple example DTMCs are presented to clarify the definitions of this section. The following four one-step transition matrices are examined for the conditions under which stationary, limiting state, and steady-state probabilities exist for them. Example 2.9 Consider the DTMC shown in Fig. 2.2 with the one-step transition probability matrix (TPM): p=

Fig. 2.2

l

l (0

O 1> *

(2.19)

Example of a discrete-time Markov chain referring to Eq. (2.19).

For this one-step TPM, an infinite number of stationary probability vectors exists: Any arbitrary probability vector is stationary in this case according to Eq. (2.15).

*The conditions under following section.

which

DTMCs

converge

to steady-state

is precisely

stated

in the

MARKOV CHAINS

l

The n-step TPM Pen) converges in the limit to: lim PC”) = fi =

n-boo

t

i

(

. >

Furthermore, all n-step TPMs are identical: P = P = Pen),

Vn E T.

The limiting state probabilities z/ do exist and are identical to the initial probability vector Y(O) in all cases: fi = Y(O)P = u(0). l

A unique steady-state probability vector does not exist for this example.

Example TPM:

2.10

Next, consider the DTMC

p=

Fig. 2.3

l

shown in Fig. 2.3 with the

O l ( 1 0> *

(2.20)

Example of a discrete-time Markov chain referring to Eq. (2.20).

For this one-step TPM, a stationary probability vector, which is unique in this case, does exist according to Eq. (2.15): u = (0.5,0.5).

l

The n-step transition matrix P cn) does not converge in the limit to any p. Therefore, the limiting state probabilities t do not exist.

l

Consequently, a unique steady-state probability

Example

2.11

Consider the DTMC shown in Fig. 2.4 with the TPM: p =

0.5 ( 0.5

l

vector does not exist.

0.5 0.5 > 4

For this one-step TPM a unique stationary probability according to Eq. (2.15): u = (0.5,0.5) .

(2.21) vector does exist

MARKOV PROCESSES

Fig. 2.4

Example of a discrete-time Markov chain referring to Eq. (2.21).

Note that this is the same unique stationary probability the different DTMC in Eq. (2.20). l

45

vector as for

The n-step TPM Pen) converges in the limit to:

Furthermore, all n-step TPMs are identical: P = @ = Pen),

Vn E T.

The limiting state probabilities fi do exist, are independent of the initial probability vector, and are unique: 2, = u(O)P = (0.5,0.5). l

A unique steady-state probability vector does exist. All probabilities are strictly positive and identical to the stationary probabilities, which can be derived from the solution of Eq. (2.15): u = ti = (0.5,0.5) .

Example

2.12

Now consider the DTMC shown in Fig. 2.5 with the TPM: (2.22)

Fig. 2.5

l

Example of a discrete-time Markov chain referring to Eq. (2.22).

For this one-step TPM a unique stationary probability according to Eq. (2.15): u = (0,l).

vector does exist

46

MARKOV CHAINS

l

The n-step TPM Pen) converges in the limit to

Furthermore, all n-step TPMs are identical: P=p=P(“),

‘inET.

The limiting state probability vector ti does exist, is independent of the initial probability vector, and is identical to the unique stationary probability vector u: ti = u = (0,l) . l

A unique steady-state probability vector does not exist for this example. The elements of the unique stationary probability vector are not strictly positive.

We proceed now to identify necessary and sufficient conditions for the existence of a steady-state probability vector of a DTMC. The conditions can be given immediately in terms of properties of the DTMC. 2.1.2.1.1 C/as.sificationsof DTMC DTMCs are categorized based on the classifications of their constituent states. Definition 2.9 Any state j is said to be reachable from any other state i, where i, j E S, if it is possible to transit from state i to state j in a finite number of steps according to the given transition probability matrix. For some integer n 2 1, the following relation must hold for the n-step transition probability: p(n) ij

>

O7

3n,n > 1.

(2.23)

A DTMC is called irreducible if all states in the chain can be reached pairwise from each other, i.e., V’i, j E S, 3n, n 2 1 : p$’ > 0. A state i E S is said to be an absorbing state5 if and only if no other state of the DTMC can be reached from it, i.e., pii = 1. Note that a DTMC containing at least one absorbing state cannot be irreducible. If countably infinite state models are encountered, we have to discriminate more accurately how states are reachable from each other. The recurrence time and the probability of recurrence must also be taken into account. 5Absorbing states play an important role in the modeling of dependable systems where transient analysis is of primary interest.

MARKOV PROCESSES

47

Let f!“’ called the n-step recurrence probability, denote Definition 2.10 the conditional probabilitj 0; the first return to state i E S in exactly n 2 1 steps after leaving state i. Then, the probability fi of ever returning to state i is given by:

fi = 2 f,!“‘.

(2.24)

n=l

Any state i E S to which the DTMC will return with probability fi = 1, is called a recurrent state; otherwise, if fi < 1, i is called a transient state. Given a recurrent state i, the mean recurrence time rni of state i of a DTMC is given by: co

rni = Cf n i(n) *

(2.25)

n=l

If the mean recurrence time is finite, that is, rni < co, i is called positive recurrent or recurrent non-null; otherwise, if rni = 00, state i is said to be recurrent null. For any recurrent state i E S, let di denote the period of state i, then di is the greatest common divisor of the set of positive integers n such that pi:) > 0. A recurrent state i is called aperiodic if its period di = 1, and periodic with period di if di > 1. It has been shown by Feller [Fe11681that the states of an irreducible DTMC are all of the same type. Hence, all states are periodic, aperiodic, transient, recurrent null, or recurrent non-null. If one of the states i of an irreducible DTMC is aperiodic Definition 2.11 then so are all the other states j E S, that is, dj = 1,Vj E S, and the DTMC itself is called aperiodic; otherwise it is said to be periodic with unique period d. An irreducible, aperiodic, discrete-time Markov chain with all states i being recurrent non-null with finite mean recurrence time rni is called an ergodic Markov chain. We are now ready to summarize the main results for the classification of discrete-time Markov chains: l

The states of a finite-state, non-null.

irreducible Markov chain are all recurrent

l

Given an aperiodic DTMC, the limits fi = limn-+oo v(n) do exist.

l

For any irreducible and aperiodic DTMC, the limit ti exists and is independent of the initial probability vector Y(O).

b For an ergodicDTMC, the limit V = (GO,fi1, fiz, . . .) exists and comprises the unique steady-state probability

vector u.

MARKOV CHAINS

48

l

The steady-state probabilities vi > 0, i E S, of an ergodic Markov chain can be obtained by solving the system of linear Eq. (2.15) or, if the (finite) mean recurrence times rni are known, by exploiting the relation: 1 vi = rni ’

vi E s.

(2.26)

If finite-state DTMCs are investigated, the solution of Eq. (2.15) can be obtained by applying standard methods for the solution of linear systems. In the infinite case, either generating functions can be applied or special structure of the one-step transition probability matrix P(l) = P may be exploited to find the solution in closed form. An example of the latter technique is given in Section 3.1 where we investigate the important class of birth-death processes. The special (tridiagonal) structure of the matrix will allow us to derive closedform solutions for the state probabilities that are not restricted to any fixed matrix size so that the limiting state probabilities of infinite state DTMCs are captured by the closed-form formulae as well. 2.1.2.1.2 DTMC State Sojourn Times The state sojourn times - the time between state changes - play an important role in the characterization of DTMCs. Only homogeneous DTMCs are considered here. We have already pointed out that the transition behavior reflects the memoryless property, that is, it only depends on the current state and neither on the history that led to the state nor on the time already spent in the current state. At every instant of time, the probability of leaving current state i is independently given by (1 - pii) = Cizj pij. Applying this repeatedly leads to a description of a random experiment in form of a sequence of Bernoulli trials with probability of success (I- pii), where “success” denotes the event of leaving current state i. Hence, the sojourn time Ri during a single visit to state i is a geometrically distributed random variable6 with pmf:

P(Ri = k) = (1 - pi,)pfi-l,

vlc E IQ+.

(2.27)

We can therefore immediately conclude that the expected sojourn time E[Ri], that is, the mean number of time steps the process spends in state i per visit, is: (2.28)

Accordingly, the va,riance var[&] given by:

of the sojourn time per visit in state i is (2.29)

6Note that distribution,

the geometric distribution i.e., it is the only discrete

is the discrete-time equivalent of the exponential distribution with the memoryless propert,y.

MARKOV PROCESSES

49

2. I .2.2 Continuous-Time Markov Chains Continuous- and discrete-time Markov chains provide different yet related modeling paradigms, each of them having their own domain of applications. For the definition of CTMCs we refer back to the definition of general Markov processes in Eq. (2.2) and specialize it to the continuous parameter, discrete state space case. CTMCs are distinct from DTMCs in the sense that state transitions may occur at arbitrary instants of time and not merely at fixed, discrete time points, as is the case with DTMCs. Therefore, we use a subset of the set of non-negative real numbers lR$ to refer to the parameter set 7’ of a CTMC, as opposed to Ne for DTMCs: Definition 2.12 A given stochastic process {Xt : t E T} constitutes a CTMC if for arbitrary ti E lR$, with 0 = te < ti < . +. < t, < tn+i, Vn E N, and ‘dsi E S = Ne for the conditional pmf, the following relation holds: fy&,+l =

= P(&,+l

%+1 =

I xtn

=

sn,&,-~

%+1

I Xtn

=

=

%-1,

* * * Jt,

=

so>

(2.30)

4.

Similar to Eq. (2.4) for DTMCs, Eq. (2.30) expresses the Marlcow property of continuous-time Markov chains. If we further impose homogeneity, then because the exponential distribution is the only continuous-time distribution that provides the memoryless property, the state sojourn times of a CTMC are necessarily exponentially distributed. Again, the right-hand side of Eq. (2.30) is referred to as the transition probability 7 pij(u, V) of the CTMC to travel from state i to state j during the period of time [u, w), with u, w E T and u 5 u: Pij (U, v) = P(X,

= j 1 X, = i).

(2.31)

For u = w we define: Pij (UT U) =

1, i = j, 0, otherwise.

(2.32)

If the transition probabilities pii(u, V) depend only on the time difference t = w - u and not on the actual values of u and w, the simplified transition probabilities for time-homogeneous CTMC result: Pij(t)

= Pij(O, t) = P(Xu+t

= j I -&A = i) = P(Xt

= j 1 x0 = i),

VJUE T.

(2.33) Given the transition probabilities pij(u, w) and the probabilities ri(u) of the CTMC at time u, the unconditional state probabilities ~j (v), j E S of the

7Note that, as opposed to the discrete-time transition steps considered here.

case, there is no fixed, discrete number of

50

MARKOV CHAINS

process at time v can be derived: Vu,v E T (u 5 w).

?tv> = Ouija,

(2.34)

iES

With P(u, V) = [pij (u, v)] as the matrix of the transition probabilities, for any pair of states i, j E S and any time interval [u, v), u, w E T, from the parameter domain, and the vector X(U) = (71-e(u),7rr(u),~(u), . . .) of the state probabilities at any instant of time u, Eq. (2.34) can be given in vectormatrix form: n(v) = rr(u)P(u, w),

Vu, u E T (u 5 v).

(2.35)

Note that for all u E T, P(u,u) = I is the identity matrix. In the time-homogeneous case, Eq. (2.34) reduces to:

TAt)= ~P,wm iES

or in vector-matrix

= ~Pij(O&r~(O), iES

(2.36)

= 7r(O)P(OJ).

(2.37)

notation: n(t) = T(O)P(i)

Similar to the discrete-time case (Eq. (2.9)), the Chapman-Kolmogorov Eq. (2.38) for the transition probabilities of a CTMC can be derived from Eq. (2.30) by applying again the theorem of total probability: PdU~4

0 5 u < w < 21.

= ~Pik(W)P&v):

(2.38)

kES

But, unlike the discrete time case, Eq. (2.38) cannot be solved easily and used directly for computing the state probabilities. Rather, it has to be transformed into a system of differential equations which, in turn, leads us to the required results. For this purpose, we define the instantaneous transition rates qij(t) (i # j) of the CTMC traveling from state i to state j. These transition rates are related to conditional transition probabilities. Consider the period of time [t, t + At), where At is chosen such that Cj,-s qij(t)At + o(k) = 18. The non-negative, finite, continuous functions qij(t) can be shown to exist under rather general conditions. For all states i, j , i # j , we define: f&j(t)=

lim

At-+0

qii(t) = lim At-+0

pij (6 t + At)

At

,

i#j,

pii(t, t + At) - 1 At

’

(2.39) (2.40)

‘The notation o(At) is defined such that limAt+o q = 0; that is, we might substitute any function for o(At) that approaches zero faster than the linear function At.

MARKOV PROCESSES

51

If the limits do exist, it is clear from Eqs. (2.39) and (2.40) that, since c jEs pij(t, t + At) = 1, at any instant of time t:

Cqij(t)=o,

ViES.

(2.41)

jCS

The quantity -qii (t) can be interpreted as the total rate at which state i is exited (to any other state) at time t. Accordingly, q~ (t), (i # j), denotes the rate at which the CTMC leaves state i in order to transit to state j at time t. As an equivalent interpretation, we can regard qij(t)At + o(At) as the transition probability pij(t, t + At) of the Markov chain to transit from state i to state j in [t, t + At), where At is chosen appropriately. Having these definitions, we return to the Chapman-Kolmogorov Eq. (2.38). Substituting v + At for v in (2.38) and subtracting both sides of the original Eq. (2.38) from the result gives us: pd”~

v + At> - P&-d

= c

P&J,

w)[;or~j(w, v + At) - ~,z&J, v)].

(2.42)

kES

Dividing both sides of Eq. (2.42) by At, taking limat+e of the resulting quotient of differences, and letting w --+ v, we derive a differential equation, the well known Kolmogorov’s forward equation: d”i$;‘“’

= ~p&h,v)qQ(v),

0 5 u < v.

(2.43)

kES

In the homogeneous case, we let t = v - u and get from Eqs. (2.39) and (2.40) time-independent transition rates qij = qij(t),Vi, j E S, such that simpler versions of the Kolmogorov’s forward differential equation for homogeneous CTMCs result: (2.44) Instead of the forward Eq. (2.43), we can equivalently derive and use the Kolmogorov’s backward equation for further computations, both in the homogeneous and non-homogeneous cases, by letting w -+ u in Eq. (2.42) and taking limat+o to get: 8Pij (u, v) dU

=

~pkj(u,+hk(U), kES

0 5 u < v.

(2.45)

Differentiating Eq. (2.34) on both sides gives Eq. (2 $47)) using the Kolmogorov’s (forward) Eq. (2.43) yields Eq. (2.48), and then, again, applying Eq. (2.34) to Eq. (2.49), we derive the differential equation for the uncondi-

52

MARKOV CHAINS

tional state probabilities ~j(v), ‘dj E S, at time v in Eq. (2.50): d Ci@

d~J+J) -----z

Pij(U, V)%(?-q

dv

(2.46)

dV

II

c

8PG (% 4 ~ @) i

ZZ iES c(c =

c

Pi&

c

+7kj(V)

qkj(u>

qkj (+k

n&)

(2.48)

)

kES

kES =

(2.47)

dV

iES

&ik(U, iES

v)%(u)

(2.49) (2.50)

(v).

kES

In the time-homogeneous case, a simpler version of Eq. (2.50) results by assuming t = v - ‘u and using time-independent transition rates qzj. So we get the system of differential Eqs. (2.51): dTi (4 &t = CQi.jni(t),

Vj E S,

(2.51)

iES

which is repeatedly used throughout this text. Usually, we prefer vectormatrix form rather than the notation used in Eq. (2.51). Therefore, for the homogeneous case, we define the infinitesimal generator matrix Q of the transition probability matrix P(t) = [pij(O, t)] = bij(t)] by referring to Eqs. (2.39) and (2.40). The matrix Q:

Q = [qiJ,

K j E S,

(2.52)

contains the transition rates qij from any state i to any other state j, where i # j, of a given continuous-time Markov chain. The elements qii on the main diagonal of Q are defined by qii = - Cj jfi qij. With the definition in Eq. (2.52), Eq. (2.51) can be given in vector-matrix form as:

+)

da = - &

= rr(t)Q.

(2.53)

For the sake of completeness, we include also the matrix form of the Kolmogorov differential equations in the time-homogeneous case. The Kolmogorev’s forward Eq. (2.44) can be written as: = P(t)&. The Kolmogorov’s matrix form as:

(2.54)

backward equation in the homogeneous case results in

Ii)(t) = T

= QP(t).

(2.55)

MARKOV PROCESSES

53

As in the discrete-time case, often the steady-state probability vector of a CTMC is of primary interest. The required properties of the steady-state probability vector, which is also called the equilibrium probability vector, are equivalent to the discrete time case. For all states i E S, the steady-state probabilities ri are: 1. Independent of time t 2. Independent of the initial state probability

vector ?r(O)

3. Strictly positive, 7ri > 0 4. Given as the time limits, ri = limt,, ri(t) = limt+W pji(t), of the state probabilities 7ri( t) and of the transition probabilities pji (t), respectively If existing for a given CTMC, the steady-state probabilities are independent of time, we immediately get: lim 3.2 t+oo dt

= 0

’

(2.56)

Under Condition (2.56), the differential Eq. (2.51) for determining the unconditional state probabilities resolves to a much simpler system of linear equations: Vj E S.

0 = Cqijri,

(2.57)

iES

In vector-matrix

form, we get accordingly: O=nQ.

(2.58)

In analogy to the discrete-time case, we call a CTMC for Definition 2.13 which a unique steady-state probability vector exists, an ergodic CTMC. The strictly positive steady-state probabilities can be gained by the unique solution of Eq. (2.58), when an additional normalization condition is imposedg. To express it in vector form, we introduce the unit vector 1 = [l, 1,. . . , llT so that the following relation holds: 7rl=

c

7Ti= 1.

(2.59)

iES

Another possibility for determining the steady-state probabilities ri for all states i E S of a CTMC, is to take advantage of a well-known relation ‘Note that besides the trivial solution 7r2 = 0, Vi E S, any vector, obtained by multiplying a solution of Eq. (2.58) by an arbitrary real-valued constant, would also yield a solution of Eq. (2.58).

MARKOV CHAINS

between the ;TT~ and the mean recurrence time lo Ik& < 00, that is, the mean time elapsed between two successive visits of the CTMC to state i: 1 ri = - &Jqii ’

vi E s.

(2.60)

In the time-homogeneous case, we can derive from Eq. (2.41) that for any j E S qjj = - &fj qji, and from Eq. (2.57) we get xi,ifj qijri = -qjj’rrj. Putting these together immediately yields the system of global balance equations:

c

qij7ri = 77-j c

i,i#j

qji,

VjES.

(2.61)

i,i#j

On the left-hand side of Eq. (2.61), the total flow from any other state i E S into state j is captured. On the right-hand side, the total flow out of state j into any other state i is summarized. The flows are balanced in steady state, i.e., they are in equilibrium. The conditions under which a CTMC is called ergodic are similar to those for a DTMC. Therefore, we can briefly summarize the criteria for classifying CTMCs and for characterizing their states. 2.1.2.2.1

Classifications

of CTMC

Definition 2.14 As for DTMCs, we call a CTMC irreducible if every state i is reachable from every other state j, where i, j E S; that is, Vi, j, i # j, 3 : pji(t) > 0. In oth er words, no proper subset 3 c S, 3 # S, of state space S exists, such that CjE3 ciEs,g qji = 0. Definition 2.15 An irreducible, homogeneous CTMC is called ergodic if and only if the unique steady-state probability vector 7r exists. As opposed to DTMCs, CTMCs cannot be periodic. Therefore, it can be shown that for an irreducible, homogeneous CTMC: l

The limits iii = limt+oo ni(t) = Zimt,,pji(t) exist Vi, j E S and are independent of the initial probability vector r(0). l1

l

The steady-state probability vector X, if existing, can be uniquely determined by the solution of the linear system of Eq. (2.58) constrained by normalization Condition (2.59).

l

A unique steady-state, or equilibrium, probability irreducible, homogeneous CTMC is finite.

l

The mean recurrence times AL&are finite for all states i E S, A4i < co, if the steady-state probability vector exists.

vector 7r exists, if the

loIn contrast to DTMCs, where lowercase notation is used to refer to recurrence time, uppercase notation is used for CTMCs. llThe limits do not necessarily constitute a steady-state probability vector.

MARKOV PROCESSES

55

2.1.2.2.2 CT/UC State So;ourn Times We have already mentioned that the distribution of the state sojourn times of a homogeneous CTMC must have the memoryless property. Since the exponential distribution is the only continuous distribution with this property, the random variables denoting the sojourn times, or holding times, must be exponentially distributed. Note that the same is true for the random variable referred to as the residual state holding time, that is, the time remaining until the next state change occurs. l2 Furthermore, the means of the two random variables are equal to l/(-qi;). Let the random variable IQ denote either the sojourn time or the residual time in state i, then the CDF is given by:

FRi(r) = 1 - eqzir,

(2.62)

r 2 0.

The mean value of I&, the mean sojourn time or the mean residual time, is given by:

q&l = --$

(2.63)

where qii is defined in Eq. (2.40). 2.1.2.3 Recapitulation We have introduced Markov chains and indicated their modeling power. The most important feature of homogeneous Markov chains is their unique memoryless property that makes them remarkably attractive. Both continuous- and discrete-time Markov chains have been defined and their properties discussed. The most important algorithms for computation of their state probabilities are discussed in following chapters. Different types of algorithms are related to different categories of Markov chains such as ergodic, absorbing, finite, or infinite chains. Furthermore, the algorithms can be divided into those applicable for computing the steady-state probabilities and those applicable for computing the time-dependent state probabilities. Others provide approximate solutions, often based on an implicit transformation of the state space. Typically, these methods fall into the categories of aggregation/disaggregation techniques. Note that this modeling approximation has to be discriminated from the mathematical properties of the core algorithms, which, in turn, can be numerically exact or approximate, independent of their relation to the underlying model. Typical examples include round-off errors in direct methods such as Gaussian elimination and convergence errors in iterative methods for the solution of linear systems.

12The

residual

time

is often

referred

to as the forward

recurrence

time.

56

MARKOV CHAINS

2.2

THE

MODELING

PROCESS

So far, we have limited our discussion to the mathematical foundations and properties of DTMCs and CTMCs. Since we follow an application-oriented approach, it is worthwhile to provide explicit guidelines on how to use the theoretical concepts in a practical manner. DTMCs and CTMCs are extensively used for performance and dependability analysis of computer systems, and the same fundamental techniques can be employed in different contexts. But the context strongly determines the kind of information that is meaningful in a concrete setting. Note that the context is not simply given by a plain technical system or a given configuration that is to be modeled. Additional requirements of desired qualities are explicitly or implicitly specified that need to be taken into account. As an illustrative example, consider the outage problem of computer systems. For a given configuration, there is no ideal way to represent it without considering the application context and defining respective goals of analyses, which naturally ought to be reflected in the model structure. In a real-time context, such as flight control, even the shortest outage might have catastrophic implications for the system being controlled. Therefore, an appropriate model must be very sensitive to such a (hopefully) rare event of short duration. In contrast, the total number of jobs processed or the work accomplished during flight time is probably a less important performance measure for such a safety-critical system. If we look at a transaction processing system, however, short outages are less significant for the success but throughput is of predominant importance. So it is not useful to represent effects of short interruptions, which are of less importance in these circumstances. There are cases where the right choice of measures is less obvious. Therefore, guidelines are equally useful for the practitioner and the beginner. Later in this chapter, a framework based on A4urlcov reward models (MRMs) is presented providing recipes for a selection of the right model type and the definition of an appropriate performance measure. But before going into details of model specification, the modeling process is sketched from a global view point, thereby identifying the major phases in the modeling life-cycle and the interdependencies between these phases. 2.2.1

Modeling

Life-cycle

Phases

There can be many reasons for modeling a given system. Two of the important ones are: 1. Existing systems are modeled for a better understanding, for analyses of deficiencies such as identification of potential bottlenecks, or for upgrading studies. 2. Models are used during the design of future systems in order to check

whet her requirements are met.

THE MODELING PROCESS

57

Different levels of detail are commonly entailed in the two cases. Since for projection and design support fewer details are known in advance, the models tend to be more abstract so that analytical/numerical techniques are more appropriate in these instances.

Control

Fig. 2.6

A simplified view of the modeling process.

In general, the modeling process can be structured into phases as depicted in Fig. 2.6. First, there are certain requirements to be met by the application; these have an impact on what type of model is used. At this point we use the term “requirements” in a very broad sense and use it to refer to everything related to the evaluation of accomplishment levels, be it task or system oriented. As an example, consider the earlier discussion on the significance of outage phases for the expressiveness of the model, which cannot be

MARKOV CHAINS

58

determined a priori without reference to an application context. More details related to this issue are discussed in the following sections. A high-level description of the model is the first step to be accomplished. Either information about a real computer system is used to build the model, or experiences gained in earlier modeling studies are implicitly used. Of course, this process is rather complicated and needs both modeling and system application-specific expertise. Conceptual validation of the correctness of the high-level model is accomplished in an iterative process of step-wise refinement, which is not represented in full detail in Fig. 2.6. Conceptual validation can be further refined [NaFi67] into “face validation,” where the involved experts try to reach consensus on the appropriateness of the model on the basis of dialogs, and into the “validation of the model assumptions,” where implicit or explicit assumptions are cross-checked. Some of the crucial properties of the model are checked by answering the following questions [HMTSl]: l

Is the model logically correct, complete, or overly detailed?

l

Are the distributional assumptions justified? How sensitive are the results to simplifications in the distributional assumptions?

l

Are other stochastic properties, such as independence assumptions,valid?

l

Is the model represented on the appropriate level? Are those features included that are most significant for the application context?

Note that a valid model is not necessarily complete in every respect, but it is one that includes the most significant effects in relation to the system requirements. It leaves higher-order effects out or simplifies them drastically. In order to produce a useful model, leaving out may be of the same importance as the validity of underlying stochastic assumptions. Once confidence is gained in the validity of the high-level model description, the next step is to generate a computational model. Sometimes the available solution techniques are inadequate for solving the high-level model. Some of these difficulties arise from: l

Stochastic dependencies and correlated processes.

l

Non-exponential distributions.

l

l

Effects of stiffness, where classes of events are described in a single model that occur at extremely different rates. System requirements that are too sophisticated leading to an overly complicated model and system measures that cannot be computed easily, such as dist,ribution functions capturing response times or cumulative work.

Note that even discrete-event simulation (DES) techniques might suffer from difficulties such as largeness or stiffness. For instance, it is a non-trivial

THE MODELING PROCESS

59

problem to simulate with high confidence scenarios entailing relatively rare events while others are occurring much more often. Large or complex problems, in contrast, require the input of excessively numerous parameter sets so that correspondingly numerous simulation runs need to be executed. Many techniques have been suggested to overcome some of the problems. Among the most important ones are: l

Hybrid approaches that allow the combination of different solution techniques to take advantage of and to combine their strengths. Examples are mixed simulation and analytical/numerical approaches, or the combination of fault trees, reliability block diagrams, or reliability graphs, and Markov models [STP96]. Also product-form queueing networks and stochastic petri nets or non-product-form networks can be combined. More generally, this approach can be characterized as intermingling of state-space based and non-state-space based methods [STP96].

l

Phase-type expansions are used to replace and approximate non-exponential distributions. As a trade-off, model size and computational complexity need to be taken into account.

l

Sometimes aggregation/disaggregation techniques can be used to eliminate undesirable model properties. An approach to eliminate stiffness for transient analysis is introduced and discussed in detail in Chapter 5.

l

Transformations can be applied leading from one domain of representation to another. Space, for example, is sometimes traded with time to replace one cumulative measure by another that is easier to calculate.

l

DES of rare events is achieved by artificially speeding up these events in a controlled manner and taking some correction measures afterwards. This technique is called importance sampling [NNHGSO].

Finally, the computational model may be prohibitively large due to the complexity of the problem at hand so as to preclude a direct use of the solution techniques. A low-level description is often out of the question due to the sheer size of the model and the error-prone process of creating it by hand. What is needed are techniques allowing for a compact or high-level representation of the lower-level model to be analyzed. Usually the higher-level representation languages are referred to as specification techniques and come with paradigms providing application specific structures that can be of help to practitioners. Two primary methods to deal with large models: l

Many high-level specification techniques, queueing systems, generalized stochastic Petri nets (GSPNs), and stochastic reward nets (SRNs), being the most prominent representatives, have been suggested in the literature to automate the model generation [HaTr93]. While GSPNs/SRNs are covered in more detail in Section 2.2.3, our major theme is queueing systems. Both approaches can be characterized as tolerating largeness

60

MA RKOV CHAINS

of the underlying computational for generating them. l

models and providing effective means

Another way to deal with large models is to avoid the creation of such models from the beginning. The major largeness-avoidance technique we discuss in this book is that of product-form queueing networks. The main idea is that t,he structure of the underlying CTMC allows an efficient solution that obviates the need of the generation, storage, and solution of the large state space. The second method of avoiding largeness is to separate the originally single large problem into several smaller problems and to combine submodel results into an overall solution. Both approximate and exact techniques are known for dealing with such multilevel models. The flow of information needed among submodels may be acyclic, in which case we have a hierarchical model [STP96]. If the flow of needed information is not acyclic, a fixed-point iteration may be necessary [CiTr93]. Other well-known techniques applicable for limiting model sizes are state truncations [BVDT88, GCSS86] and state lumping [NicoSO].

Ideally, the computational model is automatically generated from a highlevel description based on a formal and well-understood approach such as GSPNs or SRNs. In this case, verification is implicitly provided through the proven correctness and completeness of the techniques and tools employed. In practice, heuristics are often applied and, even worse, model generation is carried out manually. In this case, verification of the correctness of the computational model with respect to a given high-level description is of importance. But because approximations are often used, correctness is not given by a one-to-one relation between computational model and high-level description. Thus errors incurred need to be explicitly specified or bounded. Note that the distinction between a high-level description and a computational model may sometimes be conceptual when the two representations coincide. Both the issues of model generation and analytical/numerical solution methods are the main topics of this book. Once computational results have been produced, they need to be validated against data collected through measurements, if available, and against system requirements. The validation results are used for modification of the existing (high-level) model or for a proof of its validity. Of course, many iterations may be required for this important task. Note that a change in the model might ultimately result in a change in the system design. There is a strong relation between measurements and modeling. Measurement data are to be used for model validation. Furthermore, model parumeterixution relies heavily on input from measurement results. In case of a system design where the computer system does not yet exist, input usually comes from measurements of earlier studies on similar systems. Conversely, measurement studies are better planned and executed if guided by a model and by requirements placed on the computer system to be measured. In prac-

THE MODELING PROCESS

61

tice, measurements and models are not always applied on the same system. Of course, measurements cannot be used directly for the design of computer systems. Consequently, models-based solutions ought to be applied. Measurements, on the other hand, can provide the most accurate information on already existing systems. Measurement studies are often resource demanding, both with respect to hardware and software, and may require a great deal of work. The best approach, as always, is to combine the strengths of both techniques as much as possible. In the following, we examine the issue of model formulation and the formal definition of measures based on given system requirements. While the general discussion of this section is not limited to a specific model type, in what follows we narrow our view and concentrate on CMTCs. 2.2.2

Performance

Measures

We begin by introducing a simple example and then provide an introduction to Markov reward models as a means to obtain performance measures.

Meaning

Parameter l/Y 116 lb l/P C

mean time to failure mean time to repair mean time to reboot mean time to recover coverage probability

Fig. 2.7

State i E {0,1,2} RC RB

A simple model of a multiprocessor

Meaning i working processors recovery reboot

system.

2.2.2.1 A Simple Example As an example adapted from Heimann, Mittal and Trivedi [HMTSI] , consider a multiprocessor system with n processor elements processing a given workload. Each processor is subject to failures with a mean time to failure (MTTF), l/y. In case of a failure, recovery can successfully be performed with probability c. Typically, recovery takes a brief period of time with mean l/p. Sometimes, however, the system does not successfully recover from a processor failure and suffers from a more severe impact. In this case, we assume the system needs to be rebooted with longer average duration of l/a. Probability c is called the coverage factor and is usually close

62

MARKOV CHAINS

to 1. Unsuccessful recovery is most commonly caused by error propagation when the effect of a failure is not sufficiently shielded from the rest of the system. Failed processors need to be repaired, with the mean time to repair (MTTR) of l/S. Only one processor can be repaired at a time and repair of one processor does not affect the proper working of the remaining processors. If no processor is running correctly, the whole system is out of service until first repair is completed. Neither reboot nor recovery is performed when the last processor fails. If all times are assumed to be independent, exponentially distributed random variables, then a CTMC can be used to model the scenario. In Fig. 2.7 an example is given for the case of n = 2 processors and state space S = (2, RC, RB, l,O}. Since CTMC in Fig. 2.7 is ergodic, the unique steady-state probability vector z = (~2, ARC, ~TTRB, ~1, ~0) is the solution of the Eqs. (2.58) and (2.59). From Fig. 2.7 the infinitesimal generator matrix can be derived:

Q=

-27 0 o s 0 i

2cy -P o 0 0

2(1-

c)y

0 --a 0 0

Clearly, the model in Fig. 2.7 is already an abstract representation of the system. With Eqs. (2.53) and (2.58), transient and steady-state probability vectors could be computed, but it is not yet clear what kind of measures should be calculated because we have said nothing about the application context and the corresponding system requirements. We take this model as a high-level description, which may need further elaboration so that a computational model can be generated from it. We consider four classes of system requirements for the example. 1. System availability is the probability of an adequate level of service, or, in other words, the long-term fraction of time of actually delivered service. Usually, short outages can be accepted, but interruptions of longer duration or accumulated outages exceeding a certain threshold may not be tolerable. Accordingly, the model in Fig. 2.7 must be evaluated with respect to requirements from the application context. First, tolerance thresholds must be specified as to what extent total outages can be accepted. Second, the states in the model must be partitioned into two sets: one set comprising the states where the system is considered “up,” i.e., the service being actually delivered, and the complementary set comprising the states where the system is classified as “down.” In our example, natural candidates for down states are in the set: (0, RC, RI?}. But not all of them are necessarily classified as down states. Since reconfiguration generally takes a very short time, applications may well not be susceptible to such short interruptions. As a consequence, the less significant state RC could even be eliminated in the generation of the

THE MODELING PROCESS

63

computational model. Finally, the measures to obtain from the computational model must be decided. For example, the transient probability of the system being in an up state at a certain time t conditioned on

measures. 2. System reliability is the probability of uninterrupted service exceeding a certain length of time. By definition, no interruption of service at all can be tolerated. But note that it still needs to be exactly specified as to what kind of event is to be considered as an interruption. In the most restrictive application context, reconfiguration (RC) might not be acceptable. In contrast, nothing prevents the assumption to be made that even a reboot (RB) can be tolerated and only the failure of the last component leads to the single down state (0). As a third alternative, reconfiguration may be tolerated but not reboot and not the failure of the last component. The three possible scenarios are captured in Fig. 2.8. The model structures have also been adapted by introducing absorbing down states that reflect the fact that down states are considered as representing catastrophic events.

0‘2 A

0,lVariant

Variant

1

2

Variant 3 Fig. 2.8

Model variants with absorbing states capturing reliability

requirements.

3. System performance takes the capacity of different configurations into account. Typical measures to be calculated are the utilization of the resources or system throughput. Other measures of interest relate to

64

MA RKOV CHAINS

the frequency with which certain incidents occur. With respect to the model in Fig. 2.7, the individual states need to be characterized by their contribution to a successful task completion. The higher the degree of parallelism, the higher the expected accomplishment will be. But it is a non-trivial task to find the right performance indices attributable to each particular state in the computational model. The easiest way would be to assign a capacity proportional to the number of working processors. But because each such state represents a whole configuration, where each system resource and the imposed workload can have an impact on the overall performance, more accurate characterization is needed. One way would be to execute separate modeling studies for every possible configuration and to derive some more detailed measures such as statedependent effective throughputs or response time percentiles. Another way would be to expand the states and to replace some of them with a more detailed representation of the actual configuration and the workload. This approach will lead to a model of enormous size.

4. Tusk completion is reflected in the probability that a user will receive service at the required quality, or in other words, in the proportion of users being satisfied by the received service and its provided quality. Many different kinds of measures could be defined in this category. It could be the proportion of tasks being correctly processed or the probability that computation-time thresholds are not exceeded. With advanced applications such as continuous media (e.g., audio and video streams), measures are investigated relating the degree of user’s satisfaction to the quality of the delivered service. Usually, such applicationoriented measures are composed of different constituents such as timeliness, delay-variation, loss, and throughput measures. In this subsection, we have indicated how system requirements affect model formulation. Furthermore, different types of performance measures were motivated and their relation to the model representation outlined. Next, we will show how the performance measures derived from system requirements can be explicitly specified and integrated into the representation of the comput at ional model. 2.2.2.2 Markov Reward Models MRMs provide a unifying framework for an integrated specification of model structure and system requirements. We consider the explicit specification of system requirements as an essential part of the computational model. Once the model structure has been defined so that the infinitesimal generator matrix is known, the basic equations can be written depending on the given system requirements and the structure of the matrix. For the sake of completeness, we repeat here the fundamental Eqs. (2.53),

THE MODELING PROCESS

65

(2.58) and (2.59) for the analysis of CTMCs:

(t> = WQ, -d7r dt

@)

O=rQ,

= no,

7rl= 1,

where n(t) is the transient state probability vector of the CTMC and 7r is the steady-state probability vector (assuming it exists). In addition to the transient state probabilities, sometimes cumulative probabilities are of interest. Let: L(t) =

s

(2.64)

+)dw

0

then Li(t) denotes the expected total time the CTMC spends in state i during the interval [0, t). A more convenient way to calculate the cumulative state probabilities is by solution of differential Eq. (2.65) [RST89]:

$

=

L(t)Q +r(o),

L(O) = 0.

(2.65)

Closely related to the vector of cumulative state probabilities is the vector describing the time-average behavior of the CTMC [SmTrSO]: M(t) = ;L(t).

(2.66)

With I denoting the identity matrix, M(t) can be seen to satisfy the differential Eq. (2.67):

M(0) = 0.

(2.67)

With these definitions, most of the interesting measures can be defined. But the special case of models containing absorbing states, like those in Fig. 2.8, deserves additional attention. Here, it would be interesting to compute measures based on the time a CTMC spends in non-absorbing states before an absorbing state is ultimately reached. Let the state space S = A U N be partitioned into the set A of absorbing and the set N of non-absorbing (or transient) states. Then, the time spent before absorption can be calculated by taking the limit limt+oo LN(t) restricted to the states of the set N. Note that, in general, unless very special cases for the initial probability vector are considered, the limit does not exist for the states in A. Therefore, to calculate LN(oo), the initially given infinitesimal generator matrix Q is restricted to those states in N, so that matrix QN of size IN] x IN] results. Note that QN is not an infinitesimal generator matrix. Restricting also the initial probability vector ~(0) to the non-absorbing states N results in TN(O) and allows the

66

MARKOV CHAINS

computation of lim t+oo on both side of differential linear Eq. (2.68) results [STP96]:

LN(~)QN

Eq. (2.65) so that following

= -n-v(O).

(2.68)

To give an example, consider the variant three in Fig. 2.8. Here the state space S is partitioned into the set of absorbing states A = {RB, 0) and nonabsorbing states N = (2, RC, 1) so that Q reduces to:

With

LN(oo),

the mean time to absorption

MTTA

= c

(MTTA)

L&o).

can then be written

as:

(2.69)

iEN

MRMs have long been used in Markov decision theory to assign cost and reward structures to states of Markov processes for an optimization [Howa71]. Meyer [Meye80] adopted MRMs to provide a framework for an integrated approach to performance and dependability characteristics. He coined the term performability to refer to measures characterizing the ability of faulttolerant systems, that is, systems that are subject to component failures and that can perform certain tasks in the presence of failures. With MRMs, rewards can be assigned to states or to transitions between states of a CTMC. In the former case, these rewards are referred to as reward rates and in the latt’er as impulse rewards. In this text we consider state-based rewards only. The reward rates are defined based on the system requirements, be it availability, reliability, or task oriented. Let the reward rate ri be assigned to state i E S. Then, a reward ri7-i is accrued during a sojourn of time 7; in state i. Let {X(t),t 2 0) d enote a homogeneous finite-state CTMC with state space S. Then, the random variable:

w> = TX(t)

(2.70)

refers to the instantaneous reward rate of the MRM at time t. Note the difference between reward rates ri assigned to individual states i and the overall reward rate Z(t) of the MRM characterizing the stochastic process as a whole. With the instantaneous reward rate of the CTMC defined as in Eq. (2.70), the accumulated reward Y(t) in the finite time horizon [0, t) is given by:

t

t

Y(t) = 1 Z(r)&- = S,\.(,i'k. 0

0

(2.71)

THE MODELING PROCESS

67

State A Markov Reward Model 2 --

1 --

0'. t1

t2 t3

t4

Reward Rate t

0 Q =

-,“r”

-(A

0

Xl) P2

x1 --CL2

Total Reward A .

Y4 --

. . . l

y2 =y3--

.a’

yw

Yl = rot1

= 3t1

Yz=Yl+rl(t2-tl)=yl+t2-tl r

Yl --

Y3 = y2 +r2(t3

-

t2)

= y2

Y4=Y3+7$t4-t3)=y3+t4-t3

,; l

I I

I I

I I

I I

t1

t2

t3

t4

Fig. 2.9 A three-state and Y(t) processes.

,t

Markov reward model with sample paths of the X(t),

Z(t),

68

MARKOV CHAINS

For example, consider the sample paths of X(t) , Z(t) , and Y(t) processes in Fig. 2.9 adapted from [SmTrSO]. A simple three-state MRM is presented, consisting of a CTMC with infinitesimal generator matrix Q, and the reward rate vector r = (3,1,0) assigning reward rates to the states 0, 1, and 2, respectively. Assuming an initial probability vector ~(0) = (l,O, 0), the process is initiated in state 0, that is, X(0) = 0. Since the sojourn time of the first visit to state 0 is given by tl, the reward y1 = 3tl is accumulated during this period. After transition to state 1, additional reward y2 - y1 = l(t2 - tl) is earned, and so forth. While process X(t) is assumed to follow the state transition pattern as shown in Fig. 2.9, the processes Z(t) and Y(t) necessarily show the behavior as indicated, because they are depending on X(t) through the given reward vector r. While X(t) and Z(t) are discrete valued and nonmonotonic functions, Y(t), in contrast, is a continuous-valued, monotonically non-decreasing function. Based on the definitions of X(t), Z(t), and Y(t) , which are non-independent random variables, various measures can be defined. The most general measure is referred to as the performability [Meye80]: (2.72) where q(y, t) is the distribution of the accumulated reward over a finite time [0, t). Unfortunately, the performability is difficult to compute for unrestricted models and reward structures. Smaller models can be analyzed via double Laplace transform [SmTr90], while references to other algorithms are summarized in Table 2.17. The same mathematical difficulties arise if the distribution Q(y, t) of the time-average accumulated reward is to be computed:

(2.73) The problem is considerably simplified if the system requirements are limited to expectations and other moments of random variables rather t>han distribution functions of cumulative rewards. Alternatively, efficient solution algorithms are available if the rewards can be limited to a binary structure or if the model structure is acyclic. The expected instantaneous reward rate can be computed from the solution of differential Eq. (2.53): E[Z(t)]

= Cri~i(t).

(2.74)

iES

If the underlying model is ergodic, the solution of linear Eqs. (2.58) and (2.59) can be used to calculate the expected reward rate in the limit as t -+ 00:

(2.75)

THE MODELING PROCESS

69

To compute the first moment of the performability, or the expected accumulated reward, advantage can be taken of the solution of differential Eq. (2.65): E[Y(t)]

(2.76)

= Cr&(t). iES

For models with absorbing states, the limit as t --+ 00 of the expected accumulated reward exists and is called the expected accumulated reward until absorption. It follows from the solution of linear Eq. (2.68) that: (2.77)

E[Y(oo)] = Cril,(m). iEN

Also, higher moments of the random variables can be derived. As an example, the variance of the instantaneous reward rate Z(t) is obtained from: 2

var[Z(t)]

= C rf7ri(t) -

(

C ri7ri(t) iES

iES

)

.

(2.78)

Probabilities of 2(-L) and 2 can also be easily written [TMWH92]:

P[z(t)= x]= c 7ri(t),

(2.79)

iES,ri=x

P[Z=x]= c 7Ti.

(2.80)

iES,ri=X

Two different distribution functions can also be easily derived. With the transient state probabilities gained from the solution of Eq. (2.53), the probability that the instantaneous reward rate Z(t) does not exceed a certain level x is given as: P[Z(t)

5

X] =

C

ri(t).

(2.81)

The distribution P[Y(oo) 5 y] of the accumulated reward until absorption can be derived by applying a substitution according to Beaudry [Beau781 and Ciardo et al. [CMSTSO]. Th e essential idea is to generate a (computational) model with a binary reward structure and infinitesimal generator matrix Q from the given (high-level) CTMC with a general reward structure and infinitesimal generator matrix Q. This transformation is achieved by applying a normalization in the time domain according to the reward rates so that the distribution of the time until absorption of the new CTMC can be used as a substitute measure: (2.82) icN

MARKOV CHAINS

Returning to the example model in Fig. 2.7, we apply the MRM framework with respect t,o different system requirements to demonstrate how the model can be instantiated in several ways to yield meaningful computational models. The following case study [HMTSI] serves as a guideline as to how to describe models in light of different system requirements, such as availability and reliability or task-oriented and system performance. 2.2.2.3 A Case Study example in Fig. 2.7.

We now illustrate the use of MRMs by means of the

2.2.2.3.1 System Availability Availability measures are based on a binary reward structure. Assuming for the model in Fig. 2.7 that one processor is sufficient for the system to be “up” (set of states U = (2,l)) and that otherwise it is considered as being down (set of states D = {RC, RI?, 0}, where S = U U D), a reward rate 1 is attached to the states in tJ and a reward rate 0 to those in D. The resulting reward function r is summarized in Table 2.1. The instantaneous availability is given by:

A(t) = Jwwl = &R(t) iES

= -&i(t) = 7r@)+ 7rz(t), iEU

Unavailability can be calculated with a reverse reward assignment to that for availability. The instantaneous unavailability, therefore, is given by:

UA(t)= E[Z(t)]= xriri(t) = C ri(t) = ran + rRB(t)+ TO(~). iES

iED

Steady-state availability, on the other hand, is given by: A=E[Z]=Cri~i=~~i=~2+-irl. iES

iEU

The interval availability provides a time average value: A(t) = iE[Y(b)]

= Jj / Ads 0

= i E/*i(s)dr iEU 0

= i[Lz(t)

+ Ll(t)].

Note that a different classification of the model states would lead to other values of the availability measures. Instantaneous, interval, and steady-state availabilities are the fundamental measures to be applied in this context, but there are related measures that do not rely on the binary reward structure.

Other measuresrelated to availability can be considered. To compute the mean transient uptime T(l) (t) and the mean steady-state uptime Tc2)(t) in a

THE MODELING PROCESS

Tab/e 2.2 Reward for mean uptimes

Table 2.1 Reward assignment for computation of availability State i

assignment

Reward Rate r,

State i

Reward Rate rz

71

2

1

2

t

RC RB

0 0

RC RB

0 0

1 0

1 0

1 0

0

t

given time horizon [0, t), reward rates 1 in Table 2.1 are replaced by reward rates t in Table 2.2:

W(t) = +[Y(t)]

= it c

L&)

iEV

= c

L&)

= tA(t),

iEV

2d2)(t) = E[Z] = tA. Very important measures relate to the frequency of certain events of interest. Any event represented in the model can be captured here. For example, we could be interested in the average number of repair calls made in a given period of time [0, t). With assumed repair rate 6 and the fact that repair calls are made from states 0 and 1 of the model of Fig. 2.7, the transient average number of repair calls N(l)(t) and steady-state average number of repair calls M2) (t) can again be determined, with the reward function as defined in Table 2.3:

Nyt) = +[Y(t)] lm(t)

Tab/e 2.3 Reward assignment for number of repair calls in [0, t) State i

Reward Rate ri

= 6(L)(t) + L&)),

= E[Z] = tS(;lr() + 7rl).

Tab/e 2.4 Reward assignment for computation of mean number of outages exceeding certain state-dependent thresholds in finite time [O,t) with the CTMC from Fig 2.7

2

0

RC RB

0 0

State i

1 0

t6 t6

RC RB

Reward Rate r,

2

1 0

0 t&+RC tae-OrRB

72

MARKOV CHAINS

Sometimes, local tolerance levels Ti, i E S are considered such that certain events, such as outages, can be tolerated as long as they do not last longer than a specified threshold ri. In our context it would be natural to assume a tolerance with respect to duration of reconfiguration and possibly also with respect to reboot. Based on these assumptions, the reward function as shown in Table 2.4 is defined to compute the mean number of severe interruptions I(l)(t), 1c2)(t) in finite time [0, t), i.e., interruptions lasting longer than certain thresholds for the transient and the steady-state based cases, respectively: I(l)(t)

= iE[Y(t)]

Ic2) (t) = E[Z]

= PemaTRCLRC(t)

+ CliemaTRBLRB(t) + 6eehToLo(t),

= tpemPTRC7rRC + t&e-“rRB

rTTRB+ t&e-b’TOTo.

The reward rates in Table 2.4 are specified based on the observation that event duration times are exponentially distributed in CTMCs, such as the one depicted in Fig. 2.7. Therefore, if the parameter were given by X, the probability of event duration lasting longer than Ti units of time would be e--x7X. This probability is used to weight the average number of events 0. But note that the reward rates are by no means limited to exponential distributions, any known distributions could be applied here. It is useful to specify events related to timeouts or events that have a (deterministic) deadline attached to them. Again, a binary reward function T is defined that assigns reward rates 1 to up states and reward rates 0 to (absorbing) down states, as given in Table 2.5. Recalling that reliability is the likelihood that an unwanted event has not yet occured since the beginning of the system operation, reliability can be expressed as: 2.2.2.3.2

System Reliability

R(t) = P[T > t] = qqt>

= I] = 1 - P[z(t)

< o] = qqt)],

where the random variable T characterizes the time to the next occurrence of such an unwanted (failure) event. Referring to the scenarios depicted in Fig. 2.8, three different reliability functions can be identified: h(t)

= 7r2(t),

Rz(t)

= r2(t)

+ rRC(t)

+ rRB(t)

R3(t)

= x2(t)

+ rRC(t)

+ rl(t).

+ nl(t),

Remember that Ri (t) , R2 (t), and R3 (t) are computed on the basis of three different computational models. The function to be used depends on the application requirements. With a known reliability function R(t), the mean time to the occurrence of an unwanted (failure) event is given by: E[T] = MTTF = MTTA =

s

0

R(t)dt.

THE MODELING PROCESS

73

MTTF and MTTA are acronyms for mean time to failure and mean time to absorption, respectively. With the formula of Eq. (2.69), the MTTF (MTTA) can be efficiently computed. Table 2.5 Reward assignment for computation of reliability with variant-2 CTMC from Fig 2.8

Table 2.6 Reward assignment for predicting the number of catastrophic incidents with variant-2 CTMC from Fig 2.8

State i

Reward Rate pi

State i

2

1 1

RC RB 1 0

With the reliability complement: cm(t)

= 1-

2

0

1

RC RB

0 0

1 0

1 0

t-l 0

R(t) = E[Z(t)]

qqt)]

Reward Rate r,

= 1 -

given, the unreliability

23(t)

= 1 - P[T

> t] = P[T

follows as the

5

t].

It is an important observation that the unreliability is given ZLSa distribution function of a time variable. Note that we have gained this result simply by computing E[Z(t)]. The trick was to modify the computational model in the right way such that absorbing states were introduced and the appropriate reward structure was chosen, as the one depicted in Table 2.5. Rather general time distribution functions can be effectively computed in this way. This method allows the calculation of percentiles and other powerful performance measures like response-time distribution functions, which are not necessarily of exponential type [MMKT94]. It is also worth mentioning that the unreliability could be calculated based on a reward assignment complementing the one in Table 2.5. In this case, we would get UR(t) = E[Z(t)] = To(t). Related to reliability measures, we would often be interested in knowing the expected number of catastrophic events C(t) to occur in a given time interval [0, t). To this end: c(t) = ;E[Y(t)]

= y&(t)

needs to be calculated for some interesting initial up state, i E U with the reward assignment, as given in Table 2.6. In the preceding two subsections, where availability and reliability measures were considered, binary reward functions were of predominant importance. However, other reward functions have already been used to model availability- and reliability-oriented measures 2.2.2.3.3

System and Task Performance

74

MARKOV CHAlNS

such as frequencies of outages or other related incidents of interest. A more general view is presented in this section. First, we look at a particular task with some assumed task execution time x. Under the given circumstances, we would like to know the probability of successful completion of this task. Two different aspects come into the picture here: the dynamics of the system and the susceptibility of the task (or its user) to these dynamic changes in the configuration. In the case of the example in Fig 2.7, just out ages and their impact on the task completion are considered. But this situation can be generalized to different degrees of the task’s susceptibility to loss of data or other disturbing events as well. In the case of a user requiring uninterrupted service for some time x, the probabilities that no interruption occurs in the respective states are assigned as reward rates. We allow task requirement to be state dependent as well so that in state 2 the task requires x2 time units to complete and likewise for state 1. The probabilities can be easily concluded from Fig. 2.7. In state 2, for example, the probability P[O > 221 that the system will be operational for more than 22 units of time is given by e-‘YZ2. Therefore, the task interruption probability is:

conditioned on the assumption the system is running in state 2 and on the hypothesis that the first failure causes a task’s interruption. These probabilities can be assigned as reward rates as shown in Table 2.7, so that the interruption probability IP(‘) (xl, x2) of a task of length xl, x2, respectively is computed with E[Z], w hen assuming that probability of a user finding the system initially being up is given by the steady-state probabilities: IP(1)(x1,x2)

= E[Z] = (1 - e-2Y22)iTTa+ (1 - emyIc1)7rr.

Table 2.7 Reward assignment for computing a task’s interruption probability with duration x for the CTMC from Fig 2.7 State

2 RC RB 1 0

i

Reward

Rate

1 -e--2Y~2 0 0 1 _ e-Yzl

0

Table 2.8 Reward assignment for computing a task’s interruption probability in case of uncovered error or loss of required processor for the CTMC from Fig 2.7

ri State

2 RC RB 1 0

i

Reward l-

Rate

r(i)

e-(YP--c)+Y)s2

0 0 l-ee-Yzl

0

Another scenario would be that a running task is constrained to using a single processing element, even if two were available. In this case, a task would

THE MODELING PROCESS

75

only be susceptible to a failure of its own required processor or if an uncovered failure occurs that also has severe impacts on the currently running one. With the reward assignment as shown in Table 2.8 the desired interruption probability IPc2) is obtained:

,TP(2)(~l, x2) = E[Z] = (1 - e-(y(1-c)+y)z2)7r2 + (1 - e-yz1)7ri. Recall also the reward function as defined in Table 2.4. Based on these reward rates, it is also possible to compute the mean number of severe interruptions in a finite time duration. These measures can also be applied as success measures for running a task with finite execution time x and with certain interruption tolerance patterns attached to it. In particular, continuous media exhibit such a kind of susceptibility profile. Interruptions or losses can be tolerated up to a certain threshold. Beyond that threshold, a severe degradation in quality is perceived by users. Referring back to the example in Fig.2.7, it could be of interest to take into account different levels of performance that the modeled system may exhibit in various states. One simple method is to use the processing power (proportional to the number of processing elements actually working). This would lead to the reward function depicted in Table 2.9. With this reward assignment, the expected available computing capacity in steady state E[Z], the expected instantaneous capacity at time t, E[Z(t)], or the cumulative capacity deliverable in finite time [O,t), E[Y(t)], or its time average iE[Y(t)] could be computed. These availability measures are known as capacity-oriented availability. But this reward assignment is too coarse to capture performance effects accurately. First, it is overly optimistic with respect to the actual performance and, second, effects due to outages tend to be masked by the coarse reward structure and tend to become insignificant as a consequence. From this point of view, either the reward function or the model structure are ill-posed with respect to the system requirements. Table 2.9 Reward assignment for computation of capacity oriented availability measures for the CTMC from Fig 2.7

Table 2.10 Reward assignment for computing the total loss probability for the CTMC from Fig 2.7

State i

Reward Rate ri

State i

2 RC RB 1

2 0 0

2 RC RB

12

1 0

1 0

11

0

Reward Rate r(i) 1

1 1

More sophisticated approaches either take into account the user’s requirements related to the tasks to be processed or incorporate more information about the investigated system into the model. A hierarchical method is pos-

76

MARKOV CHAINS

sible, where every working configuration is modeled separately with some (queueing) models so that more interesting parameters can be derived. Trivedi et al. [TMWH92], f or example, use a finite queueing system that is subject to losses due to buffer overflow and to response time deadline violation [TSIHSO], while de M eer et al. [MTD94] apply infinite queueing systems with limits imposed on the system’s response time for processing the tasks, so that losses occur if a deadline is missed. In both cases, closed-form solutions are available to calculate the loss probabilities Zi for all working configurations or up states i E U [GrHa85]. Of course, knowledge of the processing power of the working components and of the traffic characteristics are needed to parameterize the underlying performance models. Many of the details are given in later chapters when appropriate. Table 2.11 Reward assignment for computing the normalized throughput for the CTMC from Fig 2.7

Table 2.12 Reward assignment for computing of the expected number of tasks lost in finite time for the CTMC from Fig 2.7 -

State i

State i

Reward Rate r,

2 RC RB

12x

1 0

11X x

Reward Rate r(i)

1 - 12 0 0

2 RC RB 1 0

1 -I1 0

x x

The values Zi are used to characterize the percentage loss of tasks arriving at the system in state i E U. Consequently, tasks arriving in down states are lost with a probability of 1.0, that is, Zi = 1, i E D. Furthermore, it is assumed that (quasi) steady state in the performance model is maintained in all configuration states, such as those depicted in Fig. 2.7. In other words, tasks arriving at the system in a certain state are assumed to leave the system in the same state. This assumption is admissible because the events related to traffic effects usually occur in orders of magnitude faster than events related to changes in configurations (due to failure-repair events), The resulting reward assignment is summarized in Table 2.10. Of course, a complementary reward structure as in Table 2.11 could also be chosen, so that the normalized throughput would be calculated as has been done by Meyer [MeyeBO]. The expected total loss probability, TLP, in steady state and the transient expected total loss probability TLP(t) are then given by:

7wyt) = E[Z(t)l = C liTi iEU

= /2n2(t)

+ C ITi iED + Zm(t)

+ rdt)

+ ma(t)

+ 7@(t),

THE MODELING PROCESS

77

TLP = E[Z]

= c li7ri+c hi iEU iED

= /2x2

+ llrl

+ TRC

+ TRB

+ TO.

To compute the expected number of tasks lost in finite time [0, t), two choices are available: the interval-based measure TL(r)(t) or the steady-statebased measure TLc2) (t) can be applied. In the first case, the reward rates in Table 2.10 are multiplied by the tasks’ arrival rate X to yield Table 2.12 and to compute E[Y(t)]. In the second case, E[Z] is calculated on the basis of a modified reward assignment. All reward rates from Table 2.12 need to be multiplied by the length of the investigated time horizon [O,t) to get Table 2.13. Of course, the same reward assignment could be used for the interval-based case and iE[Y(t)] could be evaluated instead:

To(t)

= E[Y(t)] =

ifU = A

iED

(z2L&>

= tX (l2r2

+ hLl@)

+ llrl

+ TRc

+ LRC(t)

+ TRB

+ LRB(t)

+ b-j(t)),

+ To).

Table 2.13 Reward assignment for computing the expected total number of tasks lost in finite time for the CTMC from Fig 2.7

Table 2.14 Reward assignment for throughput computation with the CTMC of Fig 2.10

State i

Reward Rate r,

State i

2 RC RB 1 0

t12x tX tX tllX tX

3 2 1 0

Reward Rate r, P CL I-L 0

Response time distributions and percentiles of more complicated systems can rarely be computed by simple elementary queueing systems, and closed-

78

MARKOV CHAiNS

form solutions are rarely available. Muppala et al. have shown how CTMCs with absorbing states (or their equivalent SRNs) can be exploited to compute these measures numerically in rather general cases [MuTrSl, MMKT94]. Once the probabilities that the response time exceeds a certain threshold have been derived with this method, the reward assignments can be made corresponding to Tables 2.10, 2.12, or 2.13 and the appropriate measures can be derived. The use of reward rates is not restricted to reliability, availability, and performability models. This concept can also be used in pure (failure-free) performance models to conveniently describe performance measures of interest. In many computer performance studies, expected throughput, mean response time, or utilization are the most important measures. These measures can easily be specified by means of an appropriate reward function. Throughput characterization, for example, can be achieved by assigning the state transition rate corresponding to departure from a queue (service completion) as reward rates to the state where the transition originates. Mean response time of queueing systems can be represented indirectly and computed in two steps. First, by assigning the number of customers present in a state as the reward rate to that state and to compute the measures as required. Second, it can be proven that there exists a linear correspondence between mean response time and mean number of customers present so that Little’s theorem [LittGl, King901 can be used for the computation of the mean response times from the mean number of customers. Finally, it is worth mentioning that the utilization is also based on a binary reward assignment. If a particular resource is occupied in a given state, reward rate 1 is assigned, otherwise reward rate 0 indicates the idleness of the resources. To illustrate these measures, reward definitions for a simple CTMC are considered. Imagine customers arriving at a system with exponentially distributed interarrival times with mean l/X. In the system they compete for service from a single server station. Since service is exclusively received by each customer, if more than one customer is in the system at the same time, the others have to wait in line until their turn comes. Service times are independent exponentially distributed with mean l/p. To keep the example simple, we limit the maximum number of customers in the system to three. The system is described by the MRM shown in Fig. 2.10.

Fig. 2.10 A simple CTMC with arrivals and services.

Every state in S = {3,2,1,0} tomers in the system. A state completed by the service station arrives (arc annotated with X).

of the MRM represents the number of custransition occurs if a customer’s service is (arc annotated with p) or if a new customer Tables 2.14, 2.15, and 2.16 summarize the

THE MODELING PROCESS

Table 2.15 Reward assignment for computing the mean number of customers with the CTMC of Fig 2.10 State i

Tab/e 2.16 Reward assignment for computing utilization measureswith the CTMC of Fig 2.10

Reward Rate pi

3 2 1 0

79

State i

Reward Rate rz

3 2

1 1

1

1 0

3 2 1 0

0

reward assignments for throughput, mean number of customers, and utilization measure computations, respectively. The formulae for the computation of throughput measures A(l)(t), Xc21(t), Xc3); the mean number of customers K(1)(t), K(2)(t), K(3); mean response time measures T(r) (t), Z’(2) (t), Z’(s) ; and utilization measures p(‘)(t), pc2)(t), pc3) are also given. Note that response time measures Tci) can be calculated with the help of Little’s theorem [LittGl]:

T(i) = @I.

X(‘)(t) = E[Y(t)]

= j

(c i

= CL&3(t) XC2)(Q = W(t)]

KO(t)

= E[Z(t)l

To(t)

= +(l)(t)

+ L2(t)

(2.83)

T~TQ(T)) dr = &Li(t) i

+ h(i)),

= -&n(t)

= p (r3(t)

= C

= 3n3(t)

Tini

= ; (37r3(t)

+ 7T2(t) + n1(t))

+ 27T2(t) + lTl(t>,

+ 27Q(t) + 17rl@)) )

)

80

MARKOV CHAINS

To(t)

= $w(t)

= ;

(3&(i)

h’(3) = E[Z] = &ri

T(3)

PW)

=

1&w x

=

~Lw)l

;

=

= 37r3+ 27Q + hl,

(37r3

pC3) =

5 &3(t)

E[Z]

27r2

= ; j

+ J52(t)

= c

+ 1x1) *

= x3(t)

0 =

+

= pi%(t)

,d2)(t) = +[Y(t)]

+ 2&?(t) + L&(t)),

( xri7ri(~)) i

+ n2(t)

+ 7r&),

dr = 5 C&(t) 2

+ b(t)),

rilTi=T3+7T2+7Tl.

More MRM examples with all these types of reward assignments can be found in [MTBH93], and MuTr92, and in [TMWH92] where multiprocessor architectures with different interconnection techniques are studied by defining corresponding reward functions. As another example, queues with breakdown are investigated. Tables 2.17, 2.18, 2.19, and 2.20 summarize the different reward assignments discussed in this section. 2.2.3

Generation

Methods

We have already pointed out the importance of model generation, There are many reasons for the separation of high-level model description and lower-level computational model. For one thing, it will allow the modeler to focus more on the system being modeled rather than on low-level modeling details. Recall that CTMCs and similar other state-space based models tend to become very large when real-world problems are studied. Therefore, it is attractive to be able to specify such large models in a compact way and to avoid the errorprone and tedious creation of models manually. The compact description is based on the identification of regularities or repetitive structures in the system model. Besides reducing the size of the description, such abstractions provide visual and conceptual clarity. It is worth repeating that regularities must be present in order to take advantage of automatic model generation techniques. If repetitive structures cannot be identified, the “higher-level” representation would hardly be smaller

THE MODELING PROCESS

Table 2.17

Overview

of important

Measure

Formula

Em

ci,s

ra*a

Jww1

c 2ES 7’24)

Jw WI

CiES

Jwwl

P[z(t) = Lz]

raL (t>

c aE~ rib(m) CT,=z,iES 4) c ?-$<2,iES 4) 1 - I&r %(Y> (al + uR - &)a*(~, a) = e (for small models)

81

MRM measures Literature [GaKe79], [Kuba86], [MuTrSl], [MTBH93], [TSIHSO] [BRT88], [DaBh85], [HMTSI], [MuTrSl], [NaGa88] [BRT88], [NaGa88], [MTD94], [MTBH93] [TMWH92] [Husl81], [LeWi89], [Wu82] [SoGa89] [Beau78], [CMSTSO] [STR88], [GoTa87], [CiGr87], [SoGa89], [DoIy87], [Meye82]

and may even be more cumbersome. Furthermore, since software tools, hardware resources, human labor, and time are necessary to install and run such an automatic generation facility, its application is recommended only beyond a certain model size. Finally, the structure of the high-level description language generally makes it more suited for a specific domain of applications. As an example, effects of resource sharing and queueing are naturally represented by means of queueing systems, while synchronization can be easily specified with some Petri net variant. Many high-level description languages exist, each with its own merits and limitations. A comprehensive review of specification techniques for MRMs can be found in [HaTr93]. In the remainder of this chapter, we discuss two types of stochastic Petri nets: generalized stochastic Petri nets (GSPNs) and stochastic reward nets (SRNs). We also discuss algorithms to convert a GSPN into a CTMC and an SRN into an MRM. 2.2.3.1 Petri Nets Petri nets (PNs) were originally introduced by C.A. Petri in 1962 [Petr62]. A PN is a bipartite directed graph, consisting of two types of nodes, namely places, P, and transition, 7’. Arcs between the nodes fall in two categories: input arcs lead from an input place to a transition and output arcs connect a transition to an output place. Arcs always have to lead from one type of nodes to the complementary one. A finite number of tokens can be distributed among the places and every place may contain an arbitrary (natural) number of tokens. A marking m E M is defined as a possible assignment of tokens to all places in the PN. Markings are sometimes referred to as states of the PN. If P denotes the set of places, then a marking m is amultiset,mEM ClN IpI , describing the number of tokens in each place.

MARKOV CHAINS

82

Table 2.18

Use of reward rates to compute

Requirement

Rewsr

Avazlabzlity [HMTSI],

r

[Table

1

ifiEU

0

else

measures - part I

Measure

d Assignments

L-

different

E[Z]

(steady = (instantaneous) +E[Y(t)](interval)

E[Z(t)]

2. l]

state)

P[Z(t)

= l]

Jw(~)l* 1 P[Y(=J) I T/1* Pl+w I Yl I-

Unavazlabzlaty [HMTSl], [MTD94], [MCT94]

L-

1

ifiED

0

else

E[Z](steady E[Z(t)] = (instantaneous) $E[Y(t)](interval)

state)

P[Z(t)

= l]

P[SW) 5 Yl uptzme in

Mean

I’, =

WI t) [Table

Mean

uptame in

rz =

[O,t) [Table

Approz. quency

fre-

of

calls in repair [0, t) with mean duration $ [HMTSl]

of

Frequency repair

calls

t)

[0, with duration 6 [HMTS~]

Mean age interruptzons

[HMTSl]

of

I-, =

ifiEU

0

else

r

= z

(SR:

cd)

7-t =

EL-4 (ww-ox.1 $aY(t)l

2.21 1

ifiEU

0

else

ED’-(t)1 EP’(~)I*> WY(~) 5 YI*

2. l]

tS

if i E SR

0

else

(SR: states where cd) [Table 2.31

in rnean

percentsevere

t

6

if i E SR

0

else

states where [Table 2.31

P[T 0

> pi]

EL? repair

is need-

WY(t)1 repair

is need-

if z E U W’(t)1

else

(rZ 2 0: threshold for of outage) [Table 2.41

state

i,

T: duration

*The measure is valid for models with absorbing states. N: non-absorbing states; A: absorbing states; U: up states; D: down states.

THE MODELING PROCESS

Tab/e 2.19

Use of reward rates to compute different measures - part II

Requirement

Reward

Reliabilzty

ri =

Measure

Assignments 1 0

ifiEN else

E[Z(t)], P[z(t) = 11

[Table 2.51

r2 =

Unreliability [STP96] System MTTF [HMTSl]

ri =

1 0

ifiEA else

E[Z(t)l, P[z(t) = 11

1 0

ifiEN else

Jw(=J)l*

[Table 2.51 Expected ber of trophic in [0, t) [HMTSl]

numcatasevents

r, =

ifiESc else

W’(t)1

(SC: states where catastrophe may occur, X: rate of catastrophe) [Table 2.51

Task interruption probabilzty [HMTSl], [TSIHSO]

r, =

oriCapacity ented availabilztY [HMTSl], [MTD94], [CiGr87]

r, =

Total loss probabality [MuTrSl], [TMWH92], [TSIHSO]

7-i =

Normalzzed throughput

r, =

[MeyeW, [MeyeW,

X 0

P[Oi t 0

5 xi]

if i E U else

task execution time (xi: state i, Oi: sojourn time in state i) [Table 2.7, 2.81 cap(i) 0

in

if i E U else

(cap(i): capacity [Table 2.91

1 1,

E[Zl

of state i)

ifiED else

(li: percentage [Table 2.101 l-li 0

+ W(t)1

P[z(t) = x:1,P[z(t) 5x1, PLY(t)

5 Yl

E[Zl, wZ(t)l, w(t)11 loss in state i)

$WWl, P[z(t) = xc], P[z(t) W(t) I Yl

2 xl,

ifiEU else

[Table 2.111

[GaKe79],

*The measure

E[Zl>E[Z(t)l, -W(t)l,

is valid for models with absorbing states.

P[z(t) = xl, p[z(t) 5 xl, W(t) 5 Yl

83

84

MA RKOV CHAINS

Table 2.20

Use of reward rates to compute different measures - part III

Requirement

Reward Assignments

Approx.

7-i =

total

tX1,

ifiEU else

tX

ZOSS

Measure

-w-l

[MTD94], [MeSe97]

[Table 2.131

Loss [HMTSl]

r, =

Mean number of customers [MTBH93]

ri = c(i) (c(i): number of customers in state i)

Expected throughput [MeSe97], [DoIy871, [MTBH93]

r, =

Utilization [Husl81], [GoTa87], [MTBH93]

7-i =

i

61, 6

pi 0

if i E U else

= x]

if i E ST else

service (w state i) 0 1

WI Y-wYtx T

qz(t)

rate

while

in

if resource is idle else

ww

$w(t)l~

WI, p[$W I ~1

*The measure is valid for models with absorbing states

To refer to the number of tokens at a particular place Pi, 1 5 i 5 IPI, in a marking m, the notation #(Pi,m) is used. A transition is enabled if all of its input places contain at least one token. Note that a transition having no input arc is always enabled. An enable transition can fire by removing from each input place one token and by adding to each output place one token. The firing of a transition may transform a PN from one marking into another. With respect to a given initial marking, the reachability set, RS, is defined as the set of all markings reachable through any possible firing sequences of transitions, starting from the initial marking. The 72% could be infinite if no further restrictions are imposed. Even PNs appearing as being relatively simple can give rise to large or infinite RSs. If more than one transition is simultaneously enabled but cannot fire in parallel, a conflict between transitions arises that needs to be resolved through some selection policy, for example, based on a priority scheme or according to a given pmf. Markings in which no transition is enabled are called absorbing markings.

THE MODELING PROCESS

85

For a graphical presentation, places are usually depicted as circles and transitions as bars. A simple PN with an initial marking: m0

= C&O,O,O>

y

and subsequent marking:

ml = (Lb LO>, reachable through firing of transition ti, is shown in Fig. 2.11.

p3

t. 0

t 0

p4

Fig, 2.11 A simple PN before and after the firing of transition tl S

Transition ti is enabled because its single input place Pi contains two tokens. Since P2 is empty, transition t2 is disabled. When firing takes place, one token is removed from input place PI and one token is deposited in both output places P2 and P3. In the new marking, both transitions ti and t2 are enabled. The apparent conflict between ti and t2 must be solved to decide which of the alternative (absorbing) markings will be reached: m2 = (0,2,2,0)

or

m3 = (O,O, 1,l). The reachability set in this example is given by (me, ml, m2, m3), where the markings are given as defined earlier. Definition with:

2.16

A PN is given by a 5-tuple PN = {P,T, D-, D+,mo}

l

A finite set of places P = {PI, P2, . . . , PIPI}, where each place contains a non-negative number of tokens.

l

A finite set of transitions T = {ti, . . . , +I},

l

The set of input arcs D- E (0, l}IPxTl and the set of output arcs D+ E (0, l}lPxTl. If D-(Pi,tk) = 1, then there is an input arc from place Pi to transition tk, and if D+(Pj,tl) = 1, then there is an output arc from transition tl to place Pj.

l

The initial marking

mo

E M.

such that 7’ n P = 8.

86

MARKOV CHAlNS

2.2.3.2 Generalized Stochastic Petri /Vets GSPNs generalize PNs in such a way that each transition has a jiring time assigned to it, which may be exponentially distributed or constant zero. Transitions with exponentially distributed firing times are called timed transitions, while the others are referred to as immediate transitions. The new type of transitions require enabling and firing rules to be adapted accordingly. The markings M = V U ‘T in the reachability set RS of a GSPN are partitioned into two sets, the vanishing markings v and the tangible markings 7. Vanishing markings comprise those in which at least one immediate transition is enabled. If no immediate transition is enabled, that is, only timed transitions or no transitions are enabled, a tangible marking results. Vanishing markings are not resided in for any finite non-zero time and firings are performed instantaneously in this case. Therefore, immediate transitions always have priority over timed transitions to fire. If several immediate transitions compete for firing, a specified pmf is used to break the tie. If timed transitions compete, a race model is applied so that the transition whose firing time elapses first is the next one to fire. From a given GSPN, an extended reachability graph (ERG) is generated containing the markings of the reachability set as nodes and some stochastic information attached to the arcs, thereby relating the markings to each other. Absorbing loops of vanishing markings are a priori excluded from the ERG, because a stochastic discontinuity would result otherwise. The GSPNs we are interested in are bounded, i.e., the underlying reachability set is finite. Marsan, Balbo, and Conte proved that exactly one CTMC corresponds to a given GSPN under condition that only a finite number of transitions can fire in finite time with non-zero probability [ABC84].

Fig. 2.12

A simple GSPN before and after the firing of transition tl .

Thick bars are used to represent timed transitions graphically, while thin bars are reserved for immediate transitions. An example of a GSPN is shown in Fig. 2.12. In contrast to the PN from Fig. 2.11, an exponentially distributed firing time with rate Xi has been assigned to transitions tl. While in the PN of the right part in Fig. 2.11 both transitions tl and tz were enabled, only

THE MODELING PROCESS

87

immediate transition t2 is enabled in GSPN of the right part in Fig. 2.12, because immediate transitions have priority over timed transitions. 2.2.3.3 Stochastic Reward Nets Many proposals have been launched to provide extensions to GSPNs (PNs). Some of the most prominent ones are revisited in this section: arc multiplicity, inhibitor arcs, priorities, guards, markingdependent arc multiplicity, marking-dependent firing rates, and reward rates defined at the net level. With these extensions we obtain the formalism of stochastic reward nets. We note that the first three of these extensions are already present in the GSPN formalism. Arc Multiplicity. Frequently more than one token needs to be removed from a place or deposited into a place, which can be easily represented with arc multiplicities. It could be awkward, otherwise, to draw an arc for each token to be moved. Arc multiplicity is a syntactical extension that does not increase the principal modeling power but makes the use of GSPNs more convenient. Arcs with multiplicity are represented by a small number attached to the arc or by a small line cutting through the arc. By default, a multiplicity of 1 is assumed. An example of arcs with multiplicities is depicted in Fig. 2.13.

t2

fig 2.13

Simple example of multiple arcs.

Since there are initially three tokens in place Pi, transition ti is enabled, and after elapsing of the firing time, two tokens are removed from place Pi (multiplicity 2) an d one token is deposited into place P2. No other transition is enabled in the initial marking. Since in the new marking there are two tokens in Pz and one token in PI, only t2 is enabled and fires immediately, thereby removing two tokens from P2 (multiplicity 2) and adding one token to PI. After some exponentially distributed firing time has elapsed, transition ti fires for the last time, because an absorbing marking will then be reached. Finally, there will be zero tokens in PI and one token in Pz. Inhibitor Arcs. An inhibitor arc from a place to a transition disables the transition in any marking where the number of tokens in the place is equal to or greater than the multiplicity of the inhibitor arc. An inhibitor arc exhibits a complementary semantic to the one of a corresponding ordinary arc

88

MARKOV CHAINS

(with the same multiplicity). If a transition were enabled under the conditions imposed by an ordinary arc it would never be enabled under the conditions imposed by its counterpart, an inhibitor arc, and vice versa. Graphically, an inhibitor arc is indicated by a small circle instead of an arrowhead. The use of an inhibitor arc is dernonstrated in Fig. 2.14.

Fig. 2.14

A GSPN with an inhibitor

arc having a multiplicity

assigned.

The figure entails a queue with finite capacity, which customers enter with a mean interarrival time of 1 /X as long as the queue is not full, that is, fewer than k customers are already in the system. Arrivals to a full system may not enter and are considered as lost. The arrival transition is disabled in this case. The effective arrival rate, therefore, turns out to be less than X. Note that we have made use of these kind of scenarios in earlier sections when specifying reward assignments capturing throughput losses or related performance requirements. Customers leave the system after receiving service with exponentially distributed length of mean duration l/cl. Arrival t1

41 A

Case

t2 l-

Queue1

(Priority 1)

t4

Fig. 2.15

Priorities, tions,

92

t3 -r

Queue2

ts

(Priorityz)

A GSPN with priorities.

Although inhibitor arcs can be used to specify priority relait is easier if priority assignments are explicitly introduced in the pa-

THE MODELING PROCESS

89

radigm. Priorities are specified by integer numbers assigned to transitions. A transition is enabled only if no other transition with a higher priority is enabled, In Fig. 2.15 a simple example of a GSPN with priorities is shown. An idle server is assigned with different priority to each type of waiting customers. Arriving customers belong to priority class i with probability qi, xi pi = 1. Recall that immediate transitions must always have priority

Fig. 2.16

An SRN with guards.

Guards. Guards are general predicates that determine when transitions are to be enabled. This feature provides a powerful means to simplify the graphical representation and to make GSPNs easier to understand, for example, by identifying modular substructures. With these extensions, we have moved out from GSPN class to SRNs [CBC+92]. An example for an SRN with guards is given in Fig. 2.16. A system with two types of resources being subject to failure and repair is represented. The scenario modeled captures the situation of a finite capacity queueing system with finite number of servers (net 1) and additional buffer space (net 2) usable if the active servers are temporarily occupied. The total number of customers in the system is limited by the total of failure-free buffer spots plus the failure-free server units. Hence, arriving customers, which enter with a mean interarrival time l/X (net 3), can only be admitted if fewer customers are already present than given by the current capacity. This condition is described by the guard attached to the “arrival” transition. The guard characterizing the enabling conditions of the “service” transition explicates the fact that service can only be delivered if at least one server is failure free (besides the apparent condition of the presence of at least one customer).

90

MARKOV CHAINS

1

fig. 2.17

Arrival

Place flushing modeled with marking-dependent

arc multiplicity.

Marking-Dependent Arc Multiplicity. This feature can be applied when the number of tokens to be transferred depends on the current marking. A useful application is given when places need to be “flushed” at the occurrence of a certain event, such as the appearance of a token at a particular place. Such a case is shown in Fig. 2.17. Customers arrive from outside at place Pr and are immediately forwarded and deposited at places Pz, P3, or PJ, with probabilities 41, q2, and 43, respectively. Whenever there is at least one token at place PA, transition t5 is enabled, regardless of the number of tokens in the other input places P2 and Pa. When t5 fires, one token is removed from place P4, all tokens are removed from place P2 and P3, and a single token appears in place P5. Marking-Dependent Firing Rates. A common extension of GSPNs is to allow marking-dependent firing rates, where a firing rate can be specified as a function of the number of tokens in any place of the Petri net. Restricted variants do exist, where the firing rate of a transition may depend only on the number of tokens in the input places. As an example, the firing rate of transition service in net 3 of Fig. 2.16 is a function of the number of servers currently up, that is, the number of tokens in place serv-up. Reward Rate Specification. Traditional output measures obtained from a GSPN are the throughput of a transition and the mean number of tokens in a place. Very often it is useful to have more general information such as: l

The probability enabled.

that a place Pi is empty while the transition

tj is

THE MODELING PROCESS l

The probability ously enabled.

91

that two different transitions tl and t2 are simultane-

Very general reward rates can be specified on an SRN so as to allow the user to compute such custom measures. Examples of the use of reward rates can be found in [CFMT94, MCT94, MMKT94, MaTr93, MuTr91, MuTr92, WLTV96]. 2.2.3.4 GSP/V,/SRN Analysis In this section we introduce the techniques of automated generation of stochastic processes underlying a GSPN/SRN. But it is worthwhile remembering that GSPNs can also be evaluated with discreteevent simulation techniques, a non-state-space based method, and that each technique has its merits. In general, the analysis of a GSPN/SRN can be decomposed into four subtasks:

1. Generation of the UZ&J, thereby defining the underlying stochastic process. A semi-Markov process (SMP) results, with zero or exponentially distributed sojourn times. In the case of an SRN, a reward rate for each tangible marking is also generated in this step. 2. Transformation of the SMP into a CTMC by elimination of the vanishing markings with zero sojourn times and the corresponding state transitions. 3. Steady-state, transient, or cumulative transient analyses of the CTMC with methods presented in Chapters 3, 4, and 5. 4. Computation of measures. In the the case of a GSPN, standard measures such as the average number of tokens in each place and the throughput of each timed transition are computed. For the SRN, the specification of reward rates at the net level enables the computation of very general custom measures. In the ERG, which reflects the properties of the underlying stochastic process, arcs representing the firing of timed transitions are labeled with the corresponding rates, and arcs representing the firing of immediate transitions are marked with probabilities. The ERG can be transformed into a reduced reachability graph (RG) by eliminating the vanishing markings and the corresponding transitions. Finally, a CTMC will result. In the case of an SRN, a reward vector is also generated and, hence, an MRM is produced. 2.2.3.4.1 CR&T Generation An U?$ is a directed graph with nodes corresponding to the markings (states) of a GSPN/SRN and weighted arcs representing the probabilities or rates with which marking changes occur. The basic ER&7 generation algorithm is summarized in Fig. 2.18. Starting with an initial marking mo, subsequent markings are generated by systematically following all possible firing patterns. If a new marking is

92

MARKOV CHAINS

fig. 2.18 Basic ERG generation algorithm. generated, the arcs labeled with the corresponding rates and probabilities are added to the E’JZS. The algorithm terminates if all possible markings have been generated. Three main data structures are used in this algorithm: a stack (S), the ERG, and a search structure (D). l

Stack (S): Newly generated markings are buffered in a stack until they are processed.

l

ERG: This data structure is used to represent the generated extended reachability graph.

l

Search structure (D): This data structure contains all earlier generated markings so that a current marking can be checked as to whether it is new or already included in the set D. For efficiency reasons, memory requirements and access time to the data structure should be of concern in an implementation. If the data structure is stored in hash tables, fast access results but memory requirements could be excessive. In contrast, tree structures are more flexible and can be stored more compactly, but access time could be intolerably high for unbalanced trees. Balanced trees such as AVL trees and B-trees are used for this reason [AKH97].

Elimination of Vanishing Markings The elimination of vanishing markings is an important step to be accomplished for generation of a CTMC from a given GSPN/SRN. Two different techniques can be distinguished,

2.2.3.4.2

THE MODELING PROCESS

“elimination on the fly” and “post-elimination” consider the GSPN in Fig. 2.19a. t3

p2

c

[CMTSl].

9.3

To give an example,

p4

Q t-0 1-q t&-4

Fig. 2.19 The elimination of vanishing markings demonstrated by: (a) a GSPN, (b) the underlying ER&7, (c) the resulting R&Y, and (d) the corresponding CTMC.

The underlying ERG in Fig. 2.19b contains five markings. Tangible markings are represented by oval boxes, while rectangular boxes are reserved for vanishing markings. The elimination of the vanishing markings (0, 1, 0, 0,O) and (O,O, l,O,O) leads to an RG, shown in Fig. 2.19c. The RG is a CTMC (redrawn

in Fig 2.19d).

The elimination

of vanishing

markings

can either be

integrated into the generation of the &RS (e1imination on the fly) or performed afterwards (post-elimination). The merits of both methods are discussed in the subsequent subsect ions. Consider the snapshot in Fig. 2.19b. Generation of vanishing markings such as (0, l,O, 0,O) is avoided by splitting the arc and its attached rate, which would be leading to this vanishing marking, and redirecting the resulting new arcs to subsequent markings. If at least one of these subsequent markings is also of vanishing type, the procedure must be iterated accordingly. Such a situation is indicated in Fig. 2.19b. The splitting is accomplished by weighing the rates in relation to the firing probabilities of the immediate transitions (in our example, t2 and ta). As a result, only tan2.2.3.4.3

Elimination

on the f/y

94

MARKOV CHAINS

gible markings and the arcs with adjusted rates need to be stored. Vanishing markings and the associated firing probabilities are discarded immediately. The principle of rate splitting and elimination of vanishing markings is illustrated in Fig. 2.20. rl

” T G

T

<

r2

G T

7-1 = r - q, 7-2 = r . (1 - q)

Fig. 2.20

On-the-fly elimination

of vanishing markings.

Elimination on the fly is efficient with respect to memory requirements, because vanishing markings need not be stored at all. But these savings in space have to be traded with additional cost in time. The same vanishing markings may be hit several times through the &%&generation process. The elimination step would be repeated although it had been perfectly executed before. This repetition could amount to a significant wast of execution time. 2.2.3.4.4 Post-Elimination In post-elimination, the complete &RG is stored during generation. With this technique, it is easier to recognize and resolve (transient) loops consisting of vanishing markings only [CMT90]. Furthermore, no work is duplicated during ERG generation, and, as an additional advantage, the complete information included in the ERG is kept for further use. Once the &RS has been generated, it is transformed into a CTMC by simply applying the same rate splitting method as has been described in Section 2.2.3.4.3. The success of the post-elimination method largely depends on the use of efficient matrix algorithms. Let:

pv = [PVV ] PV7] denote a matrix, which is split into transition probabilities between vanishing markings only (P”‘) and between vanishing markings and tangible markings (Pv7), where Pv is ’ of dimension ]VJ x IM 1 and the set of markings is partitioned such that M = I/ U 7. Furthermore, let:

u7 = [P

1CT]

denote a matrix, which is split into transition rates from tangible markings to vanishing markings (UT’) and between tangible markings only (UT7), where U7 is of dimension (71 x IM I. P recisely the same information as contained in the ERG is provided by U7 together with Pv. With representation in matrix form, it is easy to complete the set of transition rates by formally applying the rate-splitting method as indicated earlier.

THE MODELING PROCESS

95

The complete set of transition rates, gained after the rate-splitting method has been applied, is summarized in the rate matrix U, which is of dimension 171 x 171 [CMTSl]:

u&J77+urv

(I - PVV) --I PV?

(2.84)

Rate matrix U = [uij], which contains all initially present rates between tangible markings plus the derived rates between tangible markings, obtained by applying the rate-splitting method, needs to be completed to derive the infinitesimal generator matrix Q = [aij], the entries of which are given by: 4ij

=

ifi#j,

uij

-c

kcT,k#i

uik

(2.85)

if i = j,

where 7 denotes the set of tangible markings. Note that U may contain non-zero entries on its diagonal, which can be simply ignored as redundant information when creating the infinitesimal generator matrix Q as shown in Eq. (2.85). Once the generator matrix has been derived with the described method, numerical solution methods can be applied to yield performance measures as required. Finally, it may be of interest to realize that instead of eliminating the vanishing markings, they can be preserved and the underlying semi-Markov process can be solved numerically [CMTSO, MTD94].

fig. 2.21

Simple queueing

network.

Example 2.13 A simple example is given to demonstrate the generation of an ERG by applying the algorithm from Fig. 2.18. Consider the GSPN in Fig. 2.21a that models a simple network of queues. An equivalent queueing network representation is shown in Fig. 2.21b (f or a detailed introduction to queueing networks see Chapter 6). The network contains K = 2 customers, which are assumed to be initially in node 1. Let (i, j) denote a state of the network, where i represents the number of customers at node 1 and j those at node 2.

96

MARKOV CHAINS

The CTMC can be directly derived from the resulting &ES since there are no vanishing markings in this ERG. Example 2.14 With the example GSPN from Fig. 2.22a, the intermediate steps to create the rate matrix U and the infinitesimal generator matrix Q are demonstrated. Post-elimination is assumed so that the complete ERG is available, as indicated in Fig. 2.22b. The four matrices Pvv, Pv7, UT’, and UT7 are given by: 0> pvv _- ( 00 cl2 h-4

un-

-

I 0

pVT=

'

Pa\

0 I

'

\o 0)

( 00

0 Ql

/o

0

0

0000 0

0

UT7 = I

q4 0

!I3 0> ' o\ 0 I * CL3

\o 0 0 0)

97

THE MODELING PROCESS

/7

t

Fig. 2.22 CTMC.

P3

p3

I

n

p5

p6

(a) Simple GSPN, (b) the underlying

ERG, and (c) the corresponding

Then we get:

(I-py-1

=

Pl

( > :,

“1”

u~yI-pvv)-l=

)

uTv(I~pvv)-lpv7=

i

0

q1cL1

0 0

0 0

0

0

q2rcL1+

;

;

0

0

Q3(Q2Pl

+ P2)

t

Q4672Pl

+ P-12)

0 0 0

P2

0 0 0

)

1 i

9

98

MARKOV CHAINS

!I3k2Pl

+ P2) P3

0 0 -(w1

+ q4(q2111 +q3(q2pl

Q=

+ p2)

QPl

> 1

c?4(cIZPl

+ pa)

q3(q2/!L1 + /L2)‘

+ p2))

0 0 0

-P3

0 0

0 0 0

P3

0 0

I

With Eq. (2.85), the entries of the generator matrix Q can be determined from U. To demonstrate the automated generation of 2.2.3.5 A larger Example CTMCs from high-level GSPN description, an example of a polling system adapted from [IbTrSO] is now presented. (For an introduction to polling systems see [Taka93].) Consider a two-station single-buffer polling system as modeled by the GSPN shown in Fig. 2.23. The places in the GSPN are interpreted as follows. A token in place Pi represents the case that station 1 is idle and a token in P2 correspondingly indicates the idleness of station 2. Tokens in places P, (Ps) indicate that the server is serving station 1 (station 2). Tokens appearing in Places P3 (P4) indicate that station 1 (station 2) has generated a message. If a token appears in place P7, the server is polling station 2, while a token in place Ps indicates polling of station 1. In the situation illustrated in Fig. 2.23, both stations are idle and the server has finished polling station 1 (the server is ready to serve station 1). We assume that the time until station i generates a message is exponentially distributed with mean l/Xi. Service and polling times at station i are assumed to be exponentially distributed with mean l/pi and l/yi, respectively (i = 1,2). As soon as station 1 (station 2) generates a message, the timed transition ti (tz) fires. Consider initial marking me shown in Fig. 2.23, where Ps contains a token and P3 contains no token. In this situation the immediate transition si is enabled. It fires immediately and a token is deposited in place PT. This represents the condition that the server, when arriving at station 1 and finding no message there, immediately proceeds to poll station 2. In the case that a token is found in place P3, the timed transition t3 is enabled and its firing causes a token to be deposited in place Pi and another token to be deposited in place PT. This represents the condition that station 1 returns to the idle state while the server proceeds to poll station 2. Let Zi denote the number of tokens in place P; and rnj = (Zr, . . . , 1s) denote the jth marking of the GSPN. The ERG, obtained from the initial marking mo, is shown in Fig. 2.24. Labels attached to directed arcs represents those transitions whose firing generates the successor markings. If a previ-

THE MODELING PROCESS

Fig. 2.23

99

GSPN Model for the two-station single-buffer polling system.

ously found marking is obtained again (like ms), further generations from such a marking are not required. All sixteen possible and unique markings are included in Fig. 2.24. To reduce the clutter, several markings appear more than once in the figure. The &RG consists of four vanishing markings {mo,m4,m6,m13}, represented in the graph by rectangles and tangible markings represented by ovals. We can eliminate the vanishing markings by applying the algorithm presented in Section 2.2.3.4.2. From Fig. 2.24 we can see that all the tangible markings containing a token in place Pi indicate the condition that station 1 is idle. The set of markings in which station 1 is idle is S’(l) = { ml, m3, m7, rnlo, rnls}. The set of markings Sc2) = {ml, m2, m7, mg, rn12) gives the condition that station 2 is idle. In Fig. 2.25, the finite and irreducible CTMC corresponding to the GSPN of Fig. 2.23 is shown, where each state j corresponds to the tangible marking rnj of the &RG in Fig. 2.24. With the ergodic CTMC in Fig. 2.25, the steadystate probability vector 7r can be derived according to Eq. (2.58). Letting 7rk denote the steady-state probability of the CTMC being in state Ic and letting pci) denote the steady-state probability that station i, i = 1,2 is idle,

100

MARKOV CHAINS

Fig. 2.24

Extended reachability

graph for the GSPN Model in Fig. 2.23.

we conclude: p(l) =

Tl

p

~1+~2+~7+~9+~12.

=

+ 7r3 + r7

+ TlO + r15,

(2.86)

As can be seen in Fig. 2.23, the model is structurally symmetric relative to each station and therefore it can easily be extended to more than two stations. With Eq. (2.86), the mean number of customers E[K(i)] = Kci) at singlebuffer station i is determined as: KG) = 1 - #).

Also with Eq. (2.86), the effective arrival rate at single-buffer station Z follows aS:

101

THE MODELING PROCESS

Fig. 2.25

CTMC for the GSPN in Fig. 2.23.

From Little’s theorem [LittGl], f ormulae can be derived for the computation of mean response times of simple queueing models such as the polling system we are investigating in this section. The mean response time JY[T(~)] = T(i) at station i can be computed as:

T(i) = Idi) A;’ *

(2.87)

As an extension of the two-station model in Fig. 2.23, let us consider a symmetric polling system with n = 3,5,7,10 stations, with ,ui = ,u = 1, l/y = l/yi = 0.005, and Xi = X for all stations 1 5 i 5 n. The offered load is defined by p = x7=1 X/p = r-A/p. For a numerical computation of steady-state probabilities of CTMCs such as the one depicted in Fig. 2.25, algorithms introduced in Sections 3.3 and 3.4 can be used. In Table 2.21, the mean response times Tci) = T of single buffer schemes are given for different number of stations. For many other versions of polling system models, see [IbTr90]. With Definition 2.64, prove differential Eq. (2.65) by inteProblem 2.1 gration of Eq. (2.53) on both sides. Problem

2.2

With Definition 2.66, prove differential Eq. (2.67).

Problem

2.3

Prove linear Eq. (2.68) by observing $L~(co)

= ON.

Classify the measure mean time to absorption (MTTA) as Problem 2.4 being of transient, steady-state, or stationary type. Refer back to Eqs. (2.65) and (2.68) and give reasons for your decision. exist if the underlying model has at Does lim t+coJw(t)] Problem 2.5 least one absorbing state? If so, what would this measure mean?

102

MARKOV CHAINS

Tab/e 2.21

Mean response times T at a station

P

3 Stations

5 Stations

7 Stations

10 Stations

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.07702 1.14326 1.20813 1.27122 1.33222 1.39096 1.44732 1.50126 1.55279

1.09916 1.18925 1.28456 1.38418 1.48713 1.59237 1.69890 1.80573 1.91201

1.11229 1.21501 1.32837 1.45216 1.58582 1.72836 1.87850 2.03464 2.19507

1.12651 1.24026 1.37051 1.51879 1.68618 1.87310 2.07914 2.30293 2.54220

Problem 2.6 Specify the reward assignments to model variants one and three in Fig. 2.8 for reliability computations. Compare reliability (unreliability) R(t) (UR(t)) functions Problem 2.7 from Section 2.2.2.3.2 with the computation formula of the distribution of the accumulated reward until absorption, P[Y(oo) 5 y] in Eq. (2.82) and comment on it.

3 Steady-State Solutions of Markov Chains In this chapter, we restrict ourselves to the computation of the steady-state probability vector’ of ergo&c Markov chains. Most of the literature on solution techniques of Markov chains assumes ergodicity of the underlying model. A comprehensive source on algorithms for steady-state solution techniques is the book by Stewart [Stew94]. From Eq. (2.15) and Eq. (2.58), we have v = VP and 0 = nQ, respectively, as points of departure for the study of steady-state solution techniques. Eq. (2.15) can be transformed so that: 0 = Y(P -1).

(3.1)

Therefore, both for CTMC and DTMC, a linear system of the form: O=xA

(3.2)

needs to be solved. Due to its type of entries representing the parameters of a Markov chain, matrix A is singular and it can be shown that A is of rank n - 1 for any Markov chain of size IS’] = n. It follows immediately that the resulting set of equations is not linearly independent and that one of the equations is redundant. To yield a unique, positive solution, we must impose a normalization condition on the solution x of equation 0 = xA. One way to approach the solution of Eq. (3.2) is to directly incorporate the normalization condition xl=

’ For the sake of convenience hand notation.

sometimes

1

use the term

(3.3) ‘steady-state

analysis’

as a short-

103

104

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

into the Eq. (3.2). This can be regarded as substituting one of the columns (say, the last column) of matrix A by the unit vector 1 = [I, 1,. . . , llT. With a slight abuse of notation, we denote the new matrix also by A. The resulting linear system of non-homogeneous equations is: b=xA,

b=

[O,O,..., OJ].

(3.4)

An alternative to solving Eq. (3.2) is to separately consider normalization Eq. (3.3) as an additional step in numerical computations. We demonstrate both ways when example studies are presented. It is worthwhile pointing out that for any given ergodic CTMC, a DTMC can be constructed that yields an identical steady-state probability vector as the CTMC, and vice versa. Given the generator matrix Q = [qij] of a CTMC, we can define:

P = Q/q + I,

(3.5)

where Q is chosen such that Q > max;,jcs )qij]. Setting q = maxi,jeS lqijJ should be avoided in order to assure uperiodicity of the resulting DTMC [GLT87]. Th e resulting matrix P can be used to determine the steady-state probability vector x = u, by solving u = VP and vl = 1. This method, used to reduce a CTMC to a DTMC, is called randomization or sometimes uniformixation in the literature. If, on the other hand, a transition probability matrix P of an ergodic DTMC were given, a generator matrix Q of a CTMC can be defined by: Q=P-I.

(3.6)

By solving 0 = 7rQ under the condition ~1 = 1, the desired steady-state probability vector 7r = v can be obtained. To determine the steady-state probabilities of finite Markov chains, three different approaches for the solution of a linear system of the form 0 = xA are commonly used: direct or iterative numerical methods and techniques that yield closed-form results. Both types of numerical methods have merits of their own. Whereas direct methods yield exact results2 iterative methods are generally more efficient, both in time and space. Disadvantages of iterative methods are that for some of these met hods no guarantee of convergence can be given in general and that determination of suitable error bounds for termination of the iterations is not always easy. Since iterative methods are considerably more efficient in solving Markov chains, they are commonly used for larger models. For smaller models with fewer than a few thousand states, direct methods are reliable and accurate. Though closed-form results are highly desirable, they can be obtained for only a small class of models that have some structure in their matrix.

2Modulo round-off errors resulting from finite precision arithmetic.

SYMBOLIC SOLUTION: BIRTH-DEATH PROCESS

Fig. 3.1

Birth-death

105

process.

Problem matrix.

3.1

Show that P - I has the properties of a CTMC generator

Problem

3.2

Show that Q/q+1

has the properties of a stochastic matrix.

Problem 3.3 Define a CTMC and its generator matrix Q so that the corresponding DTMC would be periodic if randomization were applied with q = rnaxi,jeS 1qijl in Eq. (3.5).

3.1

SYMBOLIC

SOLUTION:

BIRTH-DEATH

PROCESS

Birth-death processes are Markov chains where transitions are allowed only between neighboring states. We treat the continuous time case here, but analogous results for the discrete-time case are easily obtained. A one-dimensional birth-death process is shown in Fig. 3.1 and its generator matrix is shown as Eq. (3.7): --X0 Pl

-(k

+ CLl)

. ..

x2

et*

-@3+p3)

*** ..

Xl -(Xz+p2)

“=i

ii

“0”

I93

. .

!

*

...

0 0

0

X0

:

.

(3.7)

* 1

The transition rates Xk, Ic 2 0 are state dependent birth rates and ~11,L 2 1, are referred to as state dependent death rates. Assuming ergodicity, the steadystate probabilities of CTMCs of the form depicted in Fig. 3.1 can be uniquely determined from the solution of Eq. (2.58):

Solving Eq. (3.8) f or ~1, and then using this result for substitution lc = 1 in Eq. (3.9), and solving it for 7r2 yields: 7Tl =

X0 -7r0, Pl

IT2=

AoX1 plP2ro*

with

(3.10)

106

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

Eq. (3.10) together with Eq. (3.9) suggest a general solution of the following form: (3.11) Indeed, Eq. (3.11) provides the unique solution of a one-dimensional birthdeath process. Since it is not difficult to prove this hypothesis by induction, we leave this as an exercise to the reader. From the law of total probability, xi ri = 1, we get for the probability ~0 of the CTMC being in State 0: 1 7ro = l+

,F;,

go

(3.12)

=

co k-l ik

The condition for convergence of the series in the denominator of Eq. (3.12)) which is also the condition for the ergodicity of the birth-death CTMC, is: gko,

Vk > k. :

AI,

-

< 1.

Eq. (3.11) and Eq. (3.12) are used extensively in Chapter 6 to determine the probabilities n!, for many different queueing systems. These probabilities are then used to calculate performance measures such as mean queue length, or mean waiting time for these queueing systems. We deal with multidimensional birth-death processes in Chapter 7. Problem 3.4 Consider a discrete-time birth-death process with birth probability ‘bi, the death probability di, and no state change probability 1 - bi - di in state i. Derive expressions for the steady-state probabilities and conditions for convergence [Triv82].

3.2

HESSENBERG

MATRIX:

NON-MARKOV’IAN

QUEUES

Section 3.1 shows that an infinite state CTMC (or DTMC) with a tridiagonal matrix structure can be solved to obtain a closed-form result. In this section we consider two other infinite state DTMCs. However, the structure is more complex so as to preclude the solution by “inspection” that we adopted in the previous section. Here we use the method of generating functions (or z-transform) to obtain a solution. The problems we tackle originate from non-Markovian queueing systems where the underlying stochastic process is Markov regenerative [Kulk96]. One popular method for the steady-state analysis of Markov regenerative processes is to apply embedding technique so as to produce an embedded DTMC from the given Markov regenerative process. An alternative approach is to use the method of supplementary variables [Hend72]. We follow the embedded DTMC approach.

HESSENBERG MATRIX: NON-MARKOVIAN

QUEUES

107

We are interested in the analysis of a queueing system, where customers arrive according to a Poisson process, so that successive interarrival times are independent, exponentially distributed random variables with parameter X. The customers experience service at a single server with the only restriction on the service time distribution being that its first two moments are finite. Order of service is first-come-first-served and there is no restriction on the size of the waiting room. We shall see later in Chapter 6 that this is the M/G/l queueing system. Characterizing the system in such a way that the whole history is summarized in a state, we need to specify the number of customers in the system plus the elapsed service time received by the current customer in service. This description results in a continuous state stochastic process that is difficult to analyze. But it is possible to identify time instants where the elapsed time is always known so they need not be explicitly represented. A prominent set of these time instants is given by the departure instants, i.e., when a customer has just completed receiving service and before the turn of the next customer has come. In this case, elapsed time is always zero. As a result, a state description given by the number of customers is sufficient. Furthermore, because the service time distribution is known and arrivals are Poissonian, the state transition probabilities can be easily computed. It is not difficult to prove that the stochastic process defined in the indicated way constitutes a DTMC. This DTMC is referred to as embedded into the more general continuous state stochastic process. Conditions can be identified under which the embedded DTMC is ergodic, i.e., a unique steady-state pmf does exist. In Section 3.2.1 we show how the steady-state probability vector of this DTMC can be computed under given constraints. Fortunately, it can be proven that the steady-state pmf of the embedded DTMC is the same as the limiting probability vector of the original non-Markovian stochastic process we started with. The proof of this fact relies on the so-called PASTA theorem, stating that “Poisson arrivals see time averages” [Wolf82]. The more difficult analysis of a stochastic process of non-Markovian type can thus be reduced to the analysis of a related DTMC, yielding the same steady-state probability vector as of the original process. 3.2.1

Non-Exponential

Service

Times

The service times are given by independent, identically distributed (i.i.d.) random variables and they are independent of the arrival process. Also, the first moment E[S] = 9 and the second moment E [S2] = s2 of the service time S must be finite. Upon completion of service, the customers leave the system. To define the embedded DTMC X = {X,; n = 0,l.. .}, the state space is chosen as the number of customers in the system (0, 1, . . .}. As time instants where the DTMC is defined, we select the departure epochs, that is, the points

108

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

in time when service is just completed and the corresponding customer leaves the system. Consider the number of customers X = {X,; n = 0, 1, . . .} left behind by a departing customer, labeled n. Then the state evolution until the next epoch n + 1, where the next customer, labeled n + 1, completes service, is probabilistically governed by the Poisson arrivals. Let the random variable Y describe the number of arrivals during a service epoch. The following one-step state transitions are then possible: X n+1

if X, > 0, if X, = 0.

X,+Y-1, =

1 Y,

(3.14)

Let c&k= P[Y = Ic] denote the probability of k arrivals in a service period with given distribution B(t). If we fix the service time at t, then Y is Poisson distributed with parameter At. To obtain the unconditional probability ak, we uncondition using the service time distribution:

cm ak =

(3.15)

.I0

Now the transition probabilities are given as: i>O, j>i--1, i = 0, j > 0.

P[X,,l = j 1x, = i] =

(3.16)

The transition probability matrix P of the embedded stochastic process, which is a Hessenberg matrix, is then given by: a0 al

a2

... (3.17)

P can be effectively created, because the arrivals are Poisson and the service time distribution B(t) is known. With P given as defined in Eq. (3.17), it is not difficult to prove that a DTMC has been defined, i.e., the Markov property from Eq. (2.2) holds. The reader should also verify that this DTMC is aperiodic, and irreducible. It is intuitively clear that State 0 is positive recurrent, if the server can keep up with arrivals, i.e., the mean service time 3 is smaller than than mean interarrival time l/X. Equivalently, the mean number of arrivals E[N] = N during mean service period ?? should be less than one [Triv82]: N=A;T
(3.18)

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109

We know from Section 2.1.2.1 that the states of an irreducible DTMC are all of the same type. Therefore, positive recurrence of state 0 implies positive recurrence of the DTMC constructed in the indicated way. A formal proof of the embedded DTMC being positive recurrent if and only if Relation (3.18) holds has been given, for example, by Cinlar [Cin175]. We also know from Section 2.1.2.1 that irreducible, aperiodic, and positive recurrent DTMCs are ergodic. Therefore, the embedded DTMC is ergodic if Relation (3.18) holds. Assuming the DTMC to be ergodic, the corresponding infinite set of global balance equations is given as: uo = vOa0+ vm, k+l

uk = uoak + c

(3.19)

via]c-i+l,

k > 1.

i=l

To compute the steady-state probability vector u of the DTMC, we use the method of generating functions wherein we also need to use the LST of the service time random variable. Let the service times distribution be denoted by B(t) and its LST B”(s) be given by: 00 B'(s)

(3.20)

eestdB(t).

= s 0

Defining generating functions of the state probabilities: G(x) = 2

VkZk,

(3.21)

k=O

and of the { aj } sequence: (3.22)

GA(X) = Fajzj, j=o from Eq. (3.19), we have:

G(x) = euk,zk = u()Eakz' + ~A~Yiak-i+lZk, k=O

k=O

k=Oi=l

or: G(x)

= uoGA(~)

+ c i=l

= Us&

c

wk-i+l~IC

k=i-1

+ CC uiajzjsiel ix1 j=O

110

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

= ‘OGA(X) + --(G(z) - VO)GA(Z), or:

G(4

x -

GA(X)

= u. bG&)

-

GA(d)

x

z

7

or: G(x)

= u.

ZGA(X)

-

GA(~)

X-GA(X)

(3.23) ’

Now, because the arrival process is Poisson, we can obtain the generating function of the {aj} sequence by first conditioning on a fixed service time and then unconditioning with the service time distribution:

GA(Z) = C ajzj= C J P[Y j=O

= j ) service = t)dB(t)z”

j=O o

coo0 = cs 0 j=o co

e~“t~&?(t)zi

O” ow c j!e

=

j=O

I(0

&qt)

-At )

(3.24)

cc

e -xt+xtzqq

= .I

0 co e-wl-")&qt)

= .I

= i-(x(1

-

2)).

Thus, the generating function for the {aj} sequence is given by the LST of the service time distribution at X(1 - z). Hence, we get an expression for the generating function of the DTMC steady-state probabilities:

G(z) = u. zB-(X(1 - ‘d> - B-(X(1 - ‘>> 2 - &(X(1 - 2)) B-(X(1 - z))(z - 1) = u” x -X(X(1 - 2)) *

(3.25)

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111

The generating function in Eq. (3.25) allows the computation of the infinite steady-state probability vector u of the DTMC embedded into a continuoustime stochastic process at the departure epochs. These epochs are given by the time instants when customers have just completed their generally distributed service period. The steady-state probabilities are obtained by repeated differentiations of the probability generating function G(x), evaluated at x = 1: (3.26) If we set z = 1 on the right hand side of Eq. (3.25) and since GA(~) = ET=, aj = 1, we have O/O. Differentiating the numerator and the denominator, as per L’Hospital’s rule, we obtain: G(1) = 1 = uo

B-(X(1 - z)) + z(-l)XK’(X(l 1+ XK’(X(1

=uow~ -

2)) + XK’(X(l1+ XB-‘(X(l-

- 2)) + XB”‘(X(1 - x))

x))[l-

- x))

z]

z))

B”(O) = ” 1 + XB*‘(O) ’ From Eq. (3.24) it can be easily shown B”(O) = 1 and -B-‘(O) we have:

= 3, so that

1 l=ve------l-X?? or: uo = 1 - xz7.

(3.27)

Taking the derivative of G(z) and setting z = 1, we get the expected number of customers in the system, E[X] = x, in steady state:

x=x9+

x2s2 2(1 - XS)’

(3.28)

Equation (3.28) is known as the Pollacxek-Khintchine formula. Remember that this formula has been derived by restricting observation to departure epochs. It is remarkable that this formula holds at random observation points in steady state, if the arrival process is Poisson. The proof is based on the PASTA theorem by Wolff, stating that “Poisson arrivals see time averages” [Wolf82]. We refrain from presenting numerical examples at this point, but refer the reader to Chapter 6 where many examples are given related to the stochastic process introduced in this section.

112

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

Problem

3.5

Specialize Eq. (3.28) above to:

(a) Exponential service time distribution (b) Deterministic

with parameter 1-1

service time with value l/p

(c) Erlang service time distribution

with mean service time l/p and Ic phases

In the notation of Chapter 6, these three cases correspond to M/M/l, and M/EI, / 1 queueing systems, respectively.

M/D/l,

Problem 3.6 Given a mean interarrival time of 1 second and a mean service time of 2 seconds, compute the steady-state probabilities for the three queueing systems M/M/l, M/D/l, and M/Ek/l to be idle. Compare the results for the three systems and comment. Show that the embedded discrete-time stochastic process Problem 3.7 X = {X,;n = O,l,. . .} defined at the departure time instants (with transition probability matrix P given by Eq. (3.17)), forms a DTMC, i.e., it satisfies the Markov property Eq. (2.2). Give a graphical representation of the DTMC defined by Problem 3.8 transition probability matrix P in Eq. (3.17). Show that the embedded DTMC X = {X,;n Problem 3.9 defined in this section is aperiodic and irreducible.

= O,l, . . .}

Consider a single-server queueing system with indepenProblem 3.10 dent, exponentially distributed service times. Furthermore, assume an arrival process with independent, identically distributed interarrival times; a general service time distribution is allowed. Service times and interarrival times are also independent. In the notation of Chapter 6, this is the GI/M/l queueing system. 1. Select a suitable state space and identify appropriate time instants where a DTMC X* should be embedded into this non-Markovian continuous state process. 2. Define a DTMC

X* = {X:; n = 0, 1, . . .} to be embedded into the non-Markovian continuous state process. In particular, define a DTMC X* by specifying state transitions by taking Eq. (3.14) as a model. Specify the transition probabilities of the DTMC X* and its transition probability matrix P*.

3.2.2

Server with

Vacations

Server vacations can be modeled as an extension of the approach presented in Section 3.2.1. In particular, the impact of vacations on the investigated

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QUEUES

113

performance parameters is of additive nature and can be derived with decomposition techniques. Non-Markovian queues with server vacations have proven to be a very useful class of models. They have been applied in a great variety of contexts [DoshSO], some of which are: l

Analysis of server brealcdowns, which may occur randomly and preempt a customer (if any) in service. Since breakdown (vacation) has priority over customer service, it is interesting to find out how the overall service capacity is affected by such breakdowns. Such insights can be provided through the analysis of queueing systems with vacations.

0 Investigation of maintenance strategies of computer, communication, or manufacturing systems. In contrast to breakdowns, which occur randomly, maintenance is usually scheduled at certain fixed intervals in order to optimize system dependability. l

Application of polling systems or cyclic server queues. Different types of polling systems that have been used include systems with exhaustive service, limited service, gated service, or some combinations thereof.

3.2.2.1 Polling Systems Because polling systems are often counted as one of the most important applications of queueing systems with server vacations, some remarks are in order here. While closed-form expressions are derived later in this section, numerical methods for the analysis of polling systems on the basis of GSPNs is covered in Section 2.2.3. The term polling comes from the polling data link control scheme in which a central computer interrogates each terminal on a multidrop communication line to find out whether it has data to transmit. The addressed terminal transmits data, and the computer examines the next terminal. Here, the server represents the computer, and a queue corresponds to a terminal. Basic polling models have recently been applied to analyze the performance of a variety of systems. In the late 195Os, a polling model with a single buffer for each queue was first used in an analysis of a problem in the British cotton industry involving a patrolling machine repairman [MMW57]. In the 196Os, polling models with two queues were investigated for the analysis of vehicleactuated traffic signal control [Newe69, NeOs69]. There were also some early studies from the viewpoint of queueing theory, apparently independent of traffic analysis [AMM65]. In the 197Os, with the advent of computer communication networks, extensive research was carried out on a polling scheme for data transfer from terminals on multidrop lines to a central computer. Since the early 198Os, the same model has been revived by [Bux81] and others for token passing schemes (e.g., token ring and token bus) in local area networks (LANs). In the current investigation of asynchronous transfer mode (ATM) for broadband ISDN (integrated services data network), cyclic scheduling is often proposed. Polling models have been applied for scheduling moving arms in secondary storage devices [CoHo86] and for resource arbitration and load

114

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

sharing in multiprocessor computers. A great number of applications exist in manufacturing systems, in transportation including moving passengers on circular and on back-and-forth routes, internal mail delivery, and shipyard loading, to mention a few. A major reason for the ubiquity of these applications is that the cyclic allocation of the server (resource) is natural and fair (since no station has to wait arbitrarily long) in many fields of engineering. The main aim of analyzing polling models is to find the message waiting time, defined as the time from the arrival of a randomly chosen message to the beginning of its service. The mean waiting time plus the mean service time is the mean message response time, which is the single most important performance measure in most computer communication systems. Another interesting characteristic is the polling cycle time, which is the time between the server’s visit to the same queue in successive cycles. Many variants and related models exist and have been studied. Due to their importance, the following list includes some polling systems of interest: 0 Single-service polling systems, in which the server serves only one message and continues to poll the next queue. l

Exhaustive-service polling systems, in which the server serves all the messages at a queue until it is empty before polling the next queue.

l

Gated-service polling systems, in which the server serves only those messages that are in the queue at the polling instant before moving to the next queue. In particular, message requests arriving after the server starts serving the queue will wait at the queue until the next time the server visits this queue.

l

Mixed exhaustive- and single-service polling systems.

l

Symmetric and asymmetric limited-service polling systems, in which the server serves at most Z(i) customers in in each service cycle at station i, with Z(i) = I for all stations i in the symmetric case.

3.2.2.2 Analysis As in Section 3.2.1, we embed a DTMC into a more general continuous-time stochastic process. Besides non-exponential service times S with its distribution given by B(t) and its LST given by B * (s), we consider vacations of duration V with distribution function C(t) and LST C”(s), and rest periods of the server of duration R, with distribution D(t) and LST D’(s). Rest periods and vacations are both given by i.i.d. random variables. Finally, the arrivals are again assumed to be Poisson. Our treatment here is based on that given in [KingSO]. If the queue is inspected and a customer is found to be present in inspection in epoch n, that is X, > 0, the customer is served according to its required service time, followed by a rest period of the server (see Fig. 3.2). Thus, after each service completion the server takes a rest before returning for inspection of the queue. If the queue is found to be empty on inspection, the server takes a

HESSENBERG MATRIX: NON-MARKOVIAN

Fig. 3.2

Phases of an M/G/l

QUEUES

115

queue with vacations.

vacation before it returns for another inspection. We embed a DTMC into this process at the inspection time instants. With E denoting the arrivals during service period including following rest period and F denoting the arrivals during a vacation, Eq. (3.14) d escribing possible state transitions from state X, at embedding epoch n to state X n+r at embedding epoch n + 1 can be restated:

if X, > 0, if X, = 0.

X,+=-l,

X n-t1

= F7

(3.29)

With ek = P[E = k] and fi = P[F = Z] d enoting the probabilities of Ic or I arrivals during the respective time periods, the transition probabilities of the embedded DTMC can be specified in analogy to Eq. (3.16):

p[&+l

= j 1x,

= i] =

The resulting transition by:

probability

fo

P=

Note that this transition the M/G/l case.

i>O, j>i-120, i = 0, j 2 0.

ej-i+17 1 fj7

matrix of the embedded DMTC is given

fl

fz

. ..’

eo el e2 . . . 0 eo el . . . 0 0 eo . . . * .. . ..

probability

(3.30)

(3.31)

matrix is also in Hessenberg form like

116

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

Let p denote the state probability vector of the embedded DTMC at inspection instants. Then its generating function is given by:

G(z) = epkzk

(3.32)

k=O

k=O

= PoG&)

+ 2

2

i=l =

po

~Gdd

-

k=O

i=l

(3.33)

p&4+12”

k&-l

(3.34)

GE(Z)

x-G&)

*

In analogy to Eq. (3.24)) the generating functions GE(Z) and GF (z) can be derived: GE(X) = ge.jzj j=o

P [E = j ) service = t]d[B(t) + D(t)],2 = 27 j=o 0

moo =cs eFxt y

j=o 0

.

d[B(t) + D(t)]2

s

XC me-xt(‘-z)d[B(t)

= ii-(x(1

+ D(t)]

- z>>o-(X(l-

x)),

(3.35) (3.36)

G&z) = C-(X(1 - z)).

With generating functions G(x), GE(Z), and GF(x) defined, and the transition probability matrix as given in Eq. (3.31), an expression for the probability generation function at inspection time instants can be derived for the server with vacation:

+) =pozc(‘(’ - ‘>>- B-(X(1 - x))D”(x(l - ‘d> x - B”(X(1 - x))D”(X(l

- 2))

*

(3.37)

Remembering G( 1) = 1 and by evaluating Eq. (3.37) at x = 1, again by differentiating the denominator and the numerator, we have: 1 - X(S + R) po = 1 - X(S + R) + XV’

(3.38)

So far, we have derived an expression for the probabilities pi at inspection time instants. But to arrive at the steady-state probabilities, we infer the state probabilities un at departure instants. Then an inspection at a time

instant immediatelyprecedingthe departure must find at least one customer present, i > 0, with probability

pi, because a departure occurred.

With

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QUEUES

117

ak = P[Y = k] d enoting again the probability of Ic arrivals in a service period, the probabilities vn, pi, and ak can be related as follows: nfl

v, =

Pi%+l-i

c

i.l

1 -Po

*

Then, the generating function of the steady-state queue length X(x)

= 2 VjXj j=o

is given by:

x(x>=fJ Xnn$r&%+1--i=fJ 2 In&an+l-i n=O =

i=ln=i--1

i-J-&

= 41

g

2

2=1

j=o

l

-Po>

,Zi+'-lpiaj

=

z(l

t

po)

gpiPi

2

2=1

j=o

ajxi

(3*3g)

(G(~-Po)GA(~.

Recalling GA(Z) = B-(X(1 - x)) and the formula for G(Z), we have: G(x)

= PO

xC”(X(1 - x)) - X(X(1 - Z))D”(X(lx - B-(X(1 - Z))D”(X(lx))

x - X(X(1 -z-B-(X(1

- x))D”(X(l-

Z))

- x))D-(X(l-

x))

1

xC”(X(1 - x)) - x x - B-(X(1x))D”(X(l - z))

x))

1

(3.40)

’

and: x(z)

= Z(l~Po)

=

Z[c-(X(1 - 2)) - l]B”(X(l Z - B-(X(1 - Z))W(X(l-

- x)) Z))

1 - XE[S + JL] [c-(x(1 - Z)) - l]B”(X(l z - B-(X(1x))D”(X(lWV1

- 2)) x)) *

(3.41)

Taking the derivative and setting x = 1, we have:

x=x9+

X2(S + R)2 ____

2(1-XS+R)

xv2 -

+ 277’

(3.42)

If there is no rest period, that is, R = 0, and if the server takes no vacation, that is, V = 0, then Eq. (3.42) re d uces to the Pollaczek-Khintchine formula

118

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

of Eq. (3.28). Furthermore, the average number of arrivals during a vacation, given by the term:

is simply added to the average number of customers that would be in the system without vacation. With respect to accumulating arrivals, the rest period can simply be considered as an extension of the service time. Problem 3.11 Give an interpretation of servers with vacation in terms of polling systems as discussed in Section 3.2.2.1. Give this interpretation for all types of polling systems enumerated in Section 3.2.2.1. Problem 3.12 How can servers with breakdown and maintenance strategies be modeled by servers with vacation as sketched in Fig. 3.2? What are the differences with polling systems? Problem 3.13 Give the ergodicity condition for an M/G/l vacation defined according to Fig. 3.2.

queue with

Problem 3.14 Derive the steady-state probabilities ve and ~1 of an M/G/l queue with vacation. Assume that the following parameters are given: Rest period is constant at 1 second, vacation is a constant 2 seconds, arrival rate X = 1, and service time distribution is Erlang Ic with k = 2 and mean l/p = 0.2 seconds. Check for ergodicity first. Use the same assumptions as specified in Problem 3.14 Problem 3.15 and, if steady state exists, compute the mean number of customers in system for the following cases: 1. M/G/l

queue with vacation according to Fig. 3.2.

2. Parameters same as in Problem 3.14 above but with rest period being constant at 0 seconds. 3. Parameters same as in Problem 3.15 part 2 above with vacation being also constant at 0 seconds.

3.3

NUMERICAL

SOLUTION:

DIRECT METHODS

The closed-form solution methods explored in Sections 3.1 and 3.2 exploited special structures of the Markov chain (or, equivalently, of its parameter matrix). For Markov chains with a more general structure, we need to resort to numerical methods. There are two broad classes of numerical methods to

solvethe linear systemsof equationsthat we are interestedin: direct methods and iterative methods.

Direct methods operate and modify the parameter

NUMERICAL SOLUTION: DIRECT METHODS

119

matrix. They use a fixed amount of computation time independent of the parameter values and there is no issue of convergence. But they are subject to fill-in of matrix entries, that is, original zero entries can become non-zeros. This makes the use of sparse storage difficult. Direct methods are also subject the accumulation of round-off errors. There are many direct methods for the solution of a system of linear equations. Some of these are restricted to certain regular structures of the parameter matrix that are of less importance for Markov chains, since these structures generally cannot be assumed in the case of a Markov chain. Among the techniques most commonly applied are the well known Gaussian elimination (GE) algorithm and, a variant thereof, Grassmann’s algorithm. The original version of the algorithm, which was published by Grassmann, Taksar, and Heyman, is usually referred to as the GTH algorithm [GTH85] and is based on a renewal argument. We introduce a newer variant by Kumar, Grassmann, and Billington [KGB871 w h ere interpretation gives rise to a simple relation to the GE algorithm. The GE algorithm suffers sometimes from numerical difficulties created by subtractions of nearly equal numbers. It is precisely this property that is avoided by the GTH algorithm and its variant through reformulations relying on regenerative properties of Markov chains. Cancellation errors are nicely circumvented in this way. 3.3.1

Gaussian

Elimination

As the point of departure for a discussion of Gaussian elimination, we refer back to Eq. (3.4). The idea of the algorithm is to transform the system of Eq. (3.43) which corresponds in matrix notation to Eq. (3.4), into an equivalent one by applying elementary operations on the parameter matrix that preserve the rank of the matrix: ao,oxo + alp1

+ . . . + an-l,oxn-1

= bo,

qm

+ . . . + an--1,1x,-l

= bl,

+ al,lxl

(3.43) aO,n-lxO + u~,~--~xI + . . . + ~~--l,~-lxn--l

=

bn-1.

As a result, an equivalent system of linear equations specified by Eq. (3.44) with a triangular matrix structure is derived, from which the desired solution x, which is identical to the solution of the original system given by Eq. (3.43), can be obtained:

(n-1)xo= b,(n-1), ao,o apl-2) ,

(0)

aO,n-lxO

(0)

+ ‘l,n-

I+...

(n-2)x1

= by-2) 7

x:o+al,l

+ af)Al n-lxn-l

(3.44) = b”? n 1’

120

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

If the system of linear equations has been transformed into a triangular structure, as indicated in Eq. (3.44), the final results can be obtained by means of a straightforward substitution process. Solving the first equation for 20, substituting the result in the second equation and solving it for ~1, and so on, finally leads to the calculation of CC,-~. Hence, the xi are recursively computed according to Eq. (3.45):

j-1

(3.45)

(n-j)

aki

k=O c aj~-j)“k,

.i =

172,***9n-

1.

,

To arrive at Eq. (3.44), an elimination procedure first needs to be performed on the original system of Eq. (3.43). Informally, the algorithm can be described as follows: first the nth equation of Eq. (3.43) is solved for x,-i, and then x,-r is eliminated from all other n- 1 equations. Next, the (n- 1)th equation is used to solve for x,-2, and, again, x,-2 is eliminated from the remaining n - 2 equations, and so forth. Finally, Eq. (3.44) results, where ai:' denotes the coefficient of xi in the (j + 1)th equation, obtained after the /Ah elimination step .3 For the sake of completeness it is pointed out that a!‘) a.i = ai >j * ‘-More formally, for the lath elimination step, i.e, the elimination of x,-k from equations j, j = n - k, n - lc - 1,. . . , 1, the (n - k + 1)th equation is to be multiplied on both sides by: (k-1)

-

an-k,j-l (k-l) an-k,n-k

(3.46)

’

and the result is added to both sides of the jth equation. The computation of the coefficients shown in the system of Eq. (3.47) and Eq. (3.48) for the lath elimination step is:

I 0, a..(k) _ZJ

,yy-l)

-

23

(k-1)

ai,n-k

j = n - k - 1, n - k - 2, , . . , 0, i=n-l,n-2 ?“‘T n - li,

aW)

n-k,j a(k-l) n-k,n-k

7

(3.47)

otherwise,

a(W b(') j

=

b("-l) j

_

bF:t)

n-kJ'

,

j

=

n

-

k -

1, n

-

k,.

aW)

n-k,n-k

3Note that i n t he system of Eq. (3.44) relation

k = n - j >_ 0 holds.

. . ,O.

(3.48)

NUMERICAL SOLUTION: DIRECT METHODS

121

In matrix notation we begin with the system of equations:

an-i,0

an-l,1

- - - an-l,n-1

After the elimination procedure, a modified system of equations results, which is equivalent to the original one. The resulting parameter matrix is in upper triangular form, where the parameters of the matrix are defined according to Eq. (3.47) and the vector representing the right-hand side of the equations according to Eq. (3.48):

(n-1) ao,o

0 9 xn-1)

0

arl-2)

ac2-3)

(A-2) %,l

* -* ***

al,,-i

0

(A-3) %,2 ap2-3) )

*..

a2,n-1

0

...

0

ao,n-i

an-l,n-1 /

(3.49)

=xu

The Gaussian elimination procedure takes advantage of elementary matrix operations that preserve the rank of the matrix. Such elementary operations correspond to interchanging of equations, multiplication of equations by a realvalued constant, and addition of a multiple of an equation to another equation. In matrix terms, the essential part of Gaussian elimination is provided by the factorization of the parameter matrix A into the components of an upper triangular matrix U and a lower triangular matrix L. The elements of matrix U resulted from the elimination procedure while the entries of L are the terms from Eq. (3.46) by which the columns of the original matrix A were multiplied

122

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

during the elimination process:

1 ape-l) > 0

=

($72)

(.$-3)

ap

aj;“,-3)

0

‘0

0

0

ao,n--1 **.. .

a$$2-3). . . .**

0

0'\

... 0

..*

0

1

0

i

1

0

al,,-i

ah

0 an-i,n-i.

(1) an-2,0 (1) an-2,n-2 an-l,0 an--l,n--l

= UL.

(1)

(1)

an-2,n-3 **

an-2,n-2 . . ,

an--l,n-2

J

an--l,+-l 1 (3.50)

As a result of the factorization of the parameter matrix A, the computation of the result vector x can split into two simpler steps: b=xA=xUL=yL. The solutions of both equations, first of: yL = b

(3.51)

for the vector of unknowns y and then finally of: xu=y

(3.52) (3.53)

for the vector of unknowns x is required. Note that the intermediate result y from Eq. (3.51), being necessary to compute the final results in Eq. (3.52), is identical to Eq. (3.49) and can readily be calculated with the formulae presented in Eq. (3.48). Since only the coefficients of matrix U are used in this computation, it is not necessary to compute and to represent explicitly the lower triangular matrix L. It is finally worth mentioning that pivoting is not necessary due to the structure of the underlying generator matrix, which is weakly diagonal dominant, since )q+ 1> qi,j, V’i, j. This property is inherited by the parameter matrices. Now the Gaussian elimination algorithm can be summarized as follows: Construct the parameter matrix A and the right-side vector b according to Eq. (3.4) as discussed in Chapter 3.

NUMERICAL SOLUTION: DIRECT METHODS

1.23

Carry out elimination steps or, equivalently, apply the standard algorithm to split the parameter matrix A into upper triangular matrix U and lower triangular matrix L such that Eq. (3.50) holds. Note that the parameters of U can be computed with the recursive formulae in Eq. (3.47) and the computation of L can be deliberately avoided. Compute the intermediate results y according to Eq. (3.51) or, equivalently, compute the intermediate results with the result from Eq. (3.53) according to Eq. (3.48). Perform the substitution to yield the final result x according to Eq. (3.52) by recursively applying the formulae shown in Eq. (3.45). Example 3.1 Consider the CTMC depicted in Fig. 3.3. This simple finite birth-death process is ergodic for any finite X and ,Q so that their unique steady-state probabilities can be computed. Since closed-form formulae have

Fig. 3.3

A simple finite birth-death

process.

been derived for this case, we can easily compute the steady-state probabilities ri as summarized in Table 3.1 with X = 1 and ,Q= 2 and with:

7ro =

1 c(> 3 i=O

x CL

2

3

lk = 1,2,3.

Table 3.1 Steady-state probabilities computed using closed-form expressions.

s

15

Alternatively, the state probabilities can be computed applying the Gaussian elimination method introduced in this section. The results from Table 3.1 can then be used to verify the correctness of the results.

124

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

First, the generator matrix Q is derived from Fig. 3.3:

0

0

2

-2

In order to include the normalization condition and to derive the parameter matrix A of the linear system to be solved, the last column of Q is replaced by the unit vector: -1

1

0

1

0

0

2

1

The resulting system of linear equations is fully specified according to Eq. (3.4), when vector b = (O,O, 0,l) is given. -;-w-7 ;.:‘;gw;,<# ~~~~~~~ With Eq. (3.47) and Eq. (3.48) in mind, matrix A is transformed into upper-triangular matrix U via intermediate steps A(l), Ac2), Ac3) = U. In parallel, vector b is transformed via b(l) and bc2) into bc3), resulting in the following sequence of matrices:

b(l) = (0, o, -2, I), 0

0

0

1

bc2) = (o,+-,,,), 0

01

0

bc3) = 0

0

+,-f,-a,l

.

01

For the sake of completeness we also present the lower triangular matrix L containing the factors from Eq. (3.46) used in the elimination steps:

It can be easily verified that: y = bc3) = (4

-;,

-2,l)

NUMERICAL SOLUTION: DIRECT METHODS

125

is the intermediate solution vector y of Eq. (3.51) with given lower triangular matrix L. The last step is the substitution according to the recursive Eq. (3.45) to yield the final results x as a solution of Eqs. (3.52) and (3.53): -- 8 17

b(“) yo x0=--&=-----&= > >

II

-- l5

=-

17

8 15’ UO,l --x0 Ul,l

-- 4 5

;8

=-3-------

--

5

l7 5

4 15’

15

1 =

-2 -2 8 ---------=-5 -5 15 x3 =

b3

a3,3

=A!?-

-14 -515

Y2

u2,2

a0,3 -X0 a3,3

-

al,3 -X1 a3,3

-

a2,3 -22 a3,3

-

uo,3 -x0 u3,3

-

u1,3 -X1 u3,3

-

u2,3 -22

=l-x()-xi-x2=1-

uo,2 -20 u2,2

-

u1,2 -21 u2,2

2 15’

-

u3,3

-

u3,3

8+4+2 15

1 = 15’

The computational complexity of the Gaussian elimination algorithm can be characterized by O(n3/3) multiplications or divisions and a storage requirement of O(n2), where n is the number of states, and hence the number of equations. Note that cancellation and rounding errors possibly induced by Gaussian elimination can adversely affect the results. This difficulty is specially relevant true if small parameters have to be dealt with, as is often the case with the analysis of large Markov chains, where relatively small state probabilities may result. For some analyses though, such as in dependability evaluation studies, we may be particularly interested in the probabilities of states that are relatively rarely entered, e.g., system down states or unsafe states. In this case, the accuracy of the smaller numbers can be of predominant interest. 3.3.2

The Grassmann

Algorithm

Grassmann’s algorithm constitutes a numerically stable variant of the Gaussian elimination procedure. The algorithm completely avoids subtractions

126

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

and it is therefore less sensitive to rounding and cancellation errors caused by the subtraction of nearly equal numbers. Grassmann’s algorithm was originally introduced for the analysis of ergodic, discrete-time Markov chains X = {X,; n = 0, 1, . . .} and was based on arguments from the theory of regenerative processes [GTH85]. A modification of the GTH algorithm has been suggested by Marsan, Meo, and de Souza e Silva [MSA96]. We follow the variant presented by Kumar et al., which allows a straightforward interpretation in terms of continuous-time Markov chains [KGB87]. We know from Eq. (2.41) that the following relation holds:

-qi,i =c Qi,j-

(3.54)

j,i#i

Furthermore, Eq. (2.57) can be suitably rearranged: n-l

--;TTiQi,i = c rjqj,i. j=O,j#i

(3.55)

Letting i = n - 1 and dividing Eq. (3.55) on both sides by qn-l,+l

yields:

n-2 -7rn-1

=

~. c j=o

%,n--1 ’ 4n-l,n-1’

This result can be used to eliminate rn-1 on the right side of Eq. (3.55): n-2

--7riqi,i =

c j=O,j#i

n-2

rjqj,i

-

iTT, qj,n-lqn-1,i c j=o

3

Qn-l,n-1

(3.56)

n-2 qj,n-lqn-1,i

=

rj

Qj,i

= j=O,j#i

_ ~ Qi,n-lQn-1,i

4n-l,n-1

(

>

’

*

Qn-l,n-1

Adding the last term of Eq. (3.56) on both sides of that equation results in an equation that can be interpreted similarly as Eq. (3.55): n-2 %,n-lqn-1,i -Ti

Qi,i (

4n-l,n-1

qj,n-lqn-1,i

ZZ >

rj = j=O,j#i

qj,i

,

4n-l,n-1

(

O
>

(3.57) With:

qj,i = qj,i _ qj7n-lqn-lTi = qj i + qf+;lqn--l,i, qn-l,n-1

’ C I=0

h-l,1

the transition rates of a new Markov chain, having one state less than the original one, are defined. Note that this elimination step, i.e., the computation

NUMERICAL SOLUTION: DIRECT METHODS

127

of x7j.i 7 is

achieved merely by adding non-negative quantities to originally nonnegative values Qj,i, j # i. Only the diagonal elements pi i and ai i are negative. It should be noted that the computation of the rates ii i on the diagonal of the reduced Markov chain can be completely avoided due to the property of Markov chains reflected in Eq. (3.54). I n order to assure the [i!jj,i] properly define a CTMC, it remains to be proven that the following equation: n-2 ‘j,i

c

=

0,

i=O

holds for all j : n-2 C

n-2 qj,i

=

C

+

i=O

i=O

C

q~~~lqn-l’i

i=”

C

=

C

qj,i

+

i=O

C qn-1,l I=0

qj,n-1

C i=O

n_Q2n-1’i

C qn-1,l l=O

n-l

n-2 =

n-2

n-2

n-2 qj,i

%,i

+

Qj,n-1

=

C

qj,i

=

0.

i=o

i=O

The new transition rates ~j,i can be interpreted as being composed of rate qj,i, describing direct transitions from state j to state i, and of rate qj,n-r describing transitions from state j via state n - 1 to state i with conditional branching probability: Qn-1,i n-2

C 4n-1,l I=0

This interpretation implies that no time is spent in (the otherwise eliminated) state n - 1. The elimination procedure is iteratively applied to the generator matrix with entries 41:’ of stepwise reduced state spaces until an upper triangular matrix results, where qi:’ denotes the matrix entries after having (n-1) on the applied elimination step Ic, 1 5 k 5 n - 1. Finally, each element qi,i main diagonal is equal to - 1. As usual, the elimination is followed by a substitution process to express the relations of the state probabilities to each other. Since the normalization condition has not been included initially, it must still be applied to yield the final state probability vector. Grassmann’s algorithm has been presented in terms of a CTMC generator matrix and, therefore, the parameter matrix must initially be properly defined:

A=

Q,

for a CTMC

P - I,

for a DTMC

128

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

j < 1,i = 1,

j

# i, 1 < j, i < 1 - 1,

j=i=l

j = l,i <'1.4 For 1 = 1,2, . . . , n - 1 do: l-1 ,,(n-1) Xl = c x ai1 * i=O

For i = 0, 1, . . . , n - 1 do:

In matrix notation, the parameter matrix A is decomposed into factors of an upper triangular matrix U and a lower triangular matrix L such that the following equations hold: O=xA=xUL. Of course, any (non-trivial) solution of 0 = XU is also a solution of the original equation 0 = xA. Therefore, there is no need to represent L explicitly. Example 3.2 Consider the CTMC presented in Fig. 3.4. It represents a finite queueing system with capacity 3, where customers arrive according to independent, exponentially distributed interarrival times with rate X as long as buffer space is available. Service is received from a single server in two consecutive phases, both phases having mean duration 1/(2~). The pair of integers (k, Z) assigned to a state i represents the number of customers k(i) in the system and the phase Z(i) of the customer currently in service. To construct the generator matrix Q, the states are ordered as indicated in Table 3.2. 4Settingthe elementson the diagonalequalto

-1 and below the diagonal equal to 0 is only included here for the sake of completeness. These assignments can be skipped in an actual implementation for efficiency reasons.

NUMERICAL SOLUTION: DIRECT METHODS

Fig. 3.4

A CTMC for a system with Erlang-2 service time distribution.

Table 3.2

(h 1) #

w9

A possible state ordering for Fig. 3.4

(1, 1)

0

(al)

1

(3,1>

(1,2)

(2,2)

(3,2)

3

4

5

6

2

Grassmann’s algorithm is applied with X = p = 1:

f

-1 0 0

1 -3 0

0 1 -3

0 0 1

0

0

0

2 0 0

0 2 0

0 0 2

-2 0 0 0

A=

\

0 2

0 0

0 0

0

2

0

0 -3 0 0

0

2

1

0

-3 0

1 -2

1 -3

0

0

0

0

0

1

0

2

0

0

0

0

1 -2 0

2

0

0 0 2 0

-3 2 0

0

0 2 0 0 -1

0

0

1

-3

1

0

-1 0 0 0 2 0

0

I

,o

1

0

0

-3

0

0

0

0

-1 1

1 -3 4 0 2 Fi

0 1 -3 2 1 ?I

0 0 1 -2 0 0

0 2 0

0 0 $

0 0 0

0

0

1

-3 0

1 21

0 ;

0

0

0

0

0

-1 I

,

$

,

129

130

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

I A(3)

0 0 0 0

y0 -3 2 0 0 0

0 1 -2 0 0 0

0g 0 0 -1 0 0

0 23 0 ; -1 0

0 0 1 0 $ -1

1 -3 ! 0 0 0 0

0 II -“3 0 0 0 0

0 01

0 i 0 0 -1 0 0

0 0 ; 0 l “1 0

0 0 0 1 0 1 -“l

-1 z

0

-3 0 0

0 0 0

0 0 0

-143.

-31

0

$

0 0 0 0 -1 63

=

0 0 0 0

-11 84

0 0 0 0 o\

A(“) =

(0

0

0 + -1

$ 0 0

0 0 0

0 0 0

-1 0 0

0 $ 0

0’ 0 1

)

1 0 21 3 0 -1 J

0 0 0 0 0

fo ; 0 -1 0 0 0 0

-“l 0 0 0

fi -1 0 0 0 0

0 0 0 0 0

0 $ -1 0 0 0

$ 0 0 -1 0 0

0 $ 0 l “1 0

0 0 1 0 1 -“l

Since Ac6) is in upper triangular form the solution of xAc6) = 0 can be easily obtained: 3 Xl = -x0 4 2 x4 = -21 3

11 1 3

x5 = -x4+

1 2 1 -25 2

x3 = -x2

x2 = 5x1

2 3

-22

26 =

+x3.

Because only six equations with seven unknowns were derived, one equation is deliberately added. For convenience, we let x0 = 2, thereby yielding the intermediate solution: x=

+--,--,-?-,4

11 11 2 5 21 12 24 3 6 24

NUMERICAL SOLUTION: DIRECT METHODS

131

With: -=- 1 n-l c i=o

12 xi

73’

and with: xi = -

Xi

n-l c i=o

the state probability

’ xi

vector 7r results:

T = (0.2192,0.1644,0.1507,0.0753,0.1096,0.1370,0.1438)

.

Measures of interest could be the average number of customers in the system E[N] = N: n-l N

=

c

k(i)7ri

= r1+

r4

+ 2(7b

+ WI)

+

3(7ra + n-6) = 1.507,

i=o the effective throughput: A, = X(x0 + nl + 7r2+ 7~~+ q,) = 0.7809, or the mean response time E[T] = ?;: N T = x = 1.9298. e

Comparing these results to the measures obtained from an analysis of the birth-death process for this queueing system assuming exponentially (rat her than Erlang) distributed service times with same parameters X = p = 1 shows that response time has decreased and the effective throughput has increased simply by reducing the variance of the service process: cEXP) = f. = 0 25 T/c 4 *’

k = 0 71) 2 73 >

&fCEXP)= (1 + 2 + 3) * a = ; = 1.5, pw e

= 3. ; = 0.75,

i$EW

= 3 4 _ 2 2’3*

The computational complexity of Grassmann’s algorithm is, similar to Gaussian elimination, O(n3/3) multiplications or divisions, and the storage

132

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

requirement is O(n2). Sparse storage techniques can be applied under some circumstances, when regular matrix structures, like those given with diagonal matrices, are preserved during computation. Although cancellation errors are being avoided with Grassmann’s algorithm, rounding errors can still occur, propagate, and accumulate during the computation. Therefore, applicability of the algorithm is also limited to medium size (around 500 states) Markov models.

3.4

NUMERICAL

SOLUTION:

ITERATIVE

METHODS

The main advantage of iterative methods over direct methods is that they preserve the sparsity the parameter matrix. Efficient sparse storage schemes and efficient sparsity-preserving algorithms can thus be used. Further advantages of iterative methods are based on the property of successive convergence to the desired solution. A good initial estimate can speed up the computation considerably. The evaluation can be terminated if the iterates are sufficiently close to the exact value, i.e., a prespecified tolerance level is not exceeded. Finally, because the parameter matrix is not altered in the iteration process, iterative methods do not suffer from the accumulation of round-off errors. The main disadvantage of iterative methods is that convergence is not always guaranteed and depending on the method, the rate of convergence is highly sensitive to the values of entries in the parameter matrix. 3.4.1

Convergence

of Iterative

Methods

Convergence is a very important issue for iterative methods that must be dealt with consciously. There are some heuristics that can be applied for choosing appropriate techniques for decisions on convergence, but no general algorithm for the selection of such a technique exists. In what follows we mention a few of the many issues to be considered with respect to convergence. A complete picture is beyond the scope of this text. A tolerance level E must be specified to provide a measure of how close the current iteration vector xck) is to the desired solution vector x. Because the desired solution vector is not known, an estimate of the error must be used to determine convergence. Some distance measures are commonly used to evaluate the current iteration vector xc’) in relation to some earlier iteration vectors x(‘), 1 < k. If the current iteration vector is “close enough” to earlier ones with respect to E, then this condition is taken as an indicator of convergence to the final result. If E were too small, convergence could become very slow or not take place at all. If E:were too large, accuracy requirements could be violated or, worse, convergence could be wrongly assumed. Some appropriate norm functions have to be applied in order to compare different iteration vectors. Many such norm functions exist, all having a different impact on speed and pattern of convergence. Size and type of the parameter

NUMERICAL SOLUTION: ITERATIVE METHODS

133

matrix should be taken into consideration for the right choice of such a norm function. Concerning the right choice of E and the norm function, it should be further noted that the components Q of the solution vector can differ by many orders of magnitude from each other in their values. An appropriate definition must take these differences into account and relate them to the accuracy requirements derived from the modeling context. 3.4.2

Power

Method

Referring to Eq. (2.15) immediately leads to a first, reliable, though sometimes slowly converging, iterative method for the computation of the steady-state probability vector of finite ergo&c Markov chains. For the sake of completeness we may mention here that for the power method to converge the transition probability, matrix P only needs to be aperiodic; irreducibility is not necessary. The power method mimics the transient behavior of the underlying DTMC until some stationary, not necessarily steady-state, convergence is reached. Therefore, it can also be exploited as a method for computing the transient state probability vector v(n) of a DTMC. Equation v = VP suggests starting with an initial guess of some probability vector Y(O) and repeatedly multiplying it by the transition probability matrix P until convergence to v is assured, with lim+, vci) = u. Due to the assumed ergodicity, or at least aperiodicity, of the underlying Markov chain, this procedure is guaranteed to converge to the desired fixed point of the unique steady-state probability vector. A single iteration step is as follows:

i -> 0,

yG+l) = &)p,

(3.58)

Hence, the iteration vector at step i is related to the initial probability via the Pi, the ith power of the transition probability matrix P:

y(i) = JO)pi 7 i 2 0.

vector

(3.59)

The power method is justified by the theory of eigenvalues and eigenvectors. The vector u can be interpreted as the left eigenvector y1 corresponding to the unit eigenvalue Xr = 1, where X1 is the dominant eigenvalue, which is the largest of all n eigenvalues, IX11 > IX21 2 . . - 2 IX,1 . Since the eigenvectors are mutually orthogonal, Y(O) can be expressed as a linear combination of all left eigenvectors of P so that the following relation holds: Y(O) = clyl

+ c2y2 + * * * + c,y,,

cjEII%,

ILjln.

(3.60)

Accordingly, after the first iteration step, the estimate Y(‘) = v(O)P can again be expressed in terms of the eigenvectors yi and eigenvalues Xi: Y(l) = CiAiyi

+ Q&y2

+ -. . + cnXnyn,

cj E R, 1 5 j 5 n.

(3.61)

cj E R,

(3.62)

Repeatedly applying this procedure yields: Jlci) = ClXZ;Yl

+ czAay2

+ * * * + CnAkyn,

1 < j 5 n.

134

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

Since all eigenvalues 1Xj ) < 1 for j 2 2, the ith power of the eigenvalues Xj, j > 2, approaches 0 in the limit, lim;,, l&Ii = 0. With respect to the unit eigenvalue [Xrl = 1 we obtain lXrli = 1, for all i 2 1. Therefore, Relation (3.63) holds and proves our assumption of the correct convergence of the power method: lim VCi) = lim {C&Q i-bee i-boo = h& CJ”,Yl

+ cZ&7Z + . . . + c,$Y,j (3.63)

= ClYl.

Only a renormalization remains to be performed to yield the final result of the steady-state probability vector u. An immediate consequence of the nature of the power method is that speed of convergence depends on the relative sizes of the eigenvalues. The closer the non-dominant eigenvalues are to 1, the slower the convergence, as can be concluded from Eq. (3.63). The second largest eigenvalue, 1x2I, is the dominating factor in this process, i.e., ,if it is close to 1, convergence will be slow. The algorithm of the power method is outlined as follows: Select Q appropriately: A=

P,

Q/q + 1; (0) * JO) = v(y),ViO),“‘7 Vn-1 > ( Select convergence criterion E, and let n = 0. Define some vector norm function f (Ilvcn), y(‘)J/), n > 1. Set convergence = falseRepeat until convergence:

-2.1

dn+l) = &)A;

I-1

IF f (Il~(~+l), ~(‘+l)li) < E, 1 -< n THEN convergence = true;

STEP 2 lYIIYl

8

n=n+l,l=li-1.

Example 3.3 Assume three customers circulating indefinitely between two service stations, alternately receiving service from both stations having exponentially distributed service times with parameters ,~r and ~2, respectively. Let (i, j) denote a possible state of the system, where i + j = 3 and i, j refer

NUMERICAL SOLUTION: ITERATIVE METHODS

Fig. 3.5

A birth-death

135

model of a two station cyclic queue with three customers.

to the number of customers at the first and the second stations, respectively. The CTMC shown in Fig. 3.5 models this two station cyclic queueing network. With ~1 = 1 and ~2 = 2 the following generator matrix Q results: -1 2

Together with an arbitrarily n(o>

1 -3

0 1

0 0

given initial probability

vector:

= (~3,0(~),~2,1(~>,~1,2(0),~0,3(0))

= (0.25,0.25,0.25,0.25), and, given Q = 3.0000003, a transition probability p =

0.6666667 0.6666666 0 0

0.3333333 0.0000001 0.6666666 0

matrix:

0 0.3333333 0.0000001 0.6666666

0 0 0.3333333 0.3333334

’

the power method can be applied, recalling Y(O) = 7r(0). Since we are dealing here with a small ergodic model, Gaussian elimination or Grassmann’s algorithm could also be applied to yield the following exact steady-state probability vector: n = (0.533,0.26~,0.133,0.6~). Using this result, the iteration vector can be compared to the exact probability vector T at intermediate steps for the purpose of demonstrating convergence as shown in Table 3.3. The results are gained by choosing E = 10s7 and by applying the following test of convergence, presuming n 2 1:

The number of iteration steps necessary to yield convergence to the exact results depends strongly on the uniformization factor q. Some examples demonstrating this dependency are summarized in Table 3.4.

136

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

Tab/e 3.3 y(O) y(l)

y(2) y(3) y(4) y(5)

yw

yW) y(30) y(3f3)

Intermediate steps and convergence of the power method 0.25 0.25 0.2777777722 0.2592592648 0.2716049333 0.2633744897 0.2671002038 0.2666741848 0.2666667970 0.2666666718

0.25 0.333333325 0.3888888778 0.4444444278 0.4691357926 0.4938271481 0.5276973339 0.5332355964 0.5333316384 0.5333332672

0.25 0.25 0.1944444556 0.1851851880 0.1604938370 0.1563786029 0.1357177959 0.1333746836 0.1333340504 0.1333333613

0.25 0.166666675 0.1388888944 0.1111111194 0.09876543704 0.08641975926 0.06948466636 0.06671553511 0.06666751412 0.06666669973

Table 3.4 Convergence of the power method as a function uniformization factor q

3.4.3

Jacobi’s

4

Iterations

3.000003 3.003 3.03 3.3 6 30

54 55 55 61 115 566

Method

The system of linear equations of interest to us is: b=xA.

(3.64)

The normalization condition may or may not be incorporated in Eq. (3.64). The parameters of both DTMC and CTMC are given by the entries of the matrix A = [aij]. The solution vector x will contain the unconditional state probabilities, or, if not yet normalized, a real-valued multiple thereof. If the normalization is incorporated, we have b = [0, 0, . . . , 0, 11, and b = 0 otherwise. Consider the jth equation from the system of Eqs. (3.64): (3.65)

bj = C aijxi. iES

Solving Eq. (3.65) for xj immediately results in: bj Xj

=

C aijxi i,i#j

.

(3.66)

ajj Any given approximate solution ji: = [go, 21, . . . , &-r] can be inserted for the variables xi, i # j, on the right side of Eq. (3.66). From these intermediate

NUMERICAL SOLUTION: ITERATIVE METHODS

137

values, better estimates of the xj on the left side of the equation may be obtained. Applying this procedure repeatedly and in parallel for all n equations results in our first iterative method. The values xc’) of the lath iteration step are computed from the values obtained from the (Ic - 1)st step for each equation independently: bj x(,“) 9

C a+~“-‘) i,i#j 7

-

by

E

s.

(3.67)

aj.i

The iteration may be started with an arbitrary initial vector x(O). The closer the initial vector is to the solution, the faster the algorithm will generally converge. Note that the equations can be evaluated in parallel, a fact that can be used as a means for computational speed-up. The method is therefore called the method of simultaneous displacement or, simply, the Jacobi method. Although the method is strikingly simple, it suffers from poor convergence and hence is rarely applied in its raw form. Splitting the matrix A = D - L - U into its constituents of the diagonal matrix D, the strictly lower-triangular matrix -L, and the strictly uppertriangular matrix -U provides a way to present the main computation step of the Jacobi method in matrix notation: xc’) - (b + x(‘-‘)

(U + L)) D-l.

(3.68)

Before the algorithm is presented, we may recall that we can deliberately incorporate the normalization condition into the parameter matrix A. Note that repeated normalization could have an adverse effect on convergence if large models are investigated, since relatively small numbers could result. Therefore, we leave it open whether to repeat normalization in each iteration step or not. Define parameter matrix A and b properly from generator matrix Q or transition probability matrix P. Choose initial vector x(O). Choose convergence criterion E. Choose some norm function f ([lx(“), x(“)/l), Ic > Z. Split parameter matrix A = D - L - U. convergence = false, and k = I = 1. Repeat until convergence: [I

[l

xck) = (b + x(lc-l) (U + L)) D-l;

. IF f (\\x’“’ - x(~-I)))) < E - THEN convergence = true

138

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

-ELSEk=k+landZ~{l,...,Ic}.

Consider two customers circulating among three service staExample 3.4 tions according to the following pattern. When a customer has received service of mean duration l/p1 at the first station, it then queues with probability ~12 at station two for service of mean duration 1/,~2, or with ~13 at station three with a mean service duration of l/ps. After completion of service at stations two or three, customers return with probability one back to station one. This scenario is captured by a CTMC with the state diagram in Fig. 3.6, where the notation (Ic, 1,m) indicates that there are Ic customers in station one, 1 customers in station two, and m customers in station three. We number the states of the CTMC of Fig. 3.6 as per Table 3.5.

Table 3.5

#

A possible state ordering according to Fig. 3.6

2

1

4

3

5

6

With PI = 1, ~2 = 2, ~3 = 3, ~12 = 0.4, and ~13 = 0.6, the generator matrix Q can be obtained:

Q=

-1 2 3 0 0 0

0.4 -3 0 2 3 0

0.6 0 -4 0 2 3

0 0.4 0 -2 0 0

0 0.6 0.4 0 -5 0

0 0 0.6 0 0 -3

To provide a controlled experiment with respect to the iteration process, we first compute the exact state probabilities in Table 3.6 with Grassmann’s algorithm introduced in Section 3.3.2. With Table 3.6 we know the exact state probabilities x = 7r and can use them for comparison with the iteration vector xc’). With the following constraints concerning equation vector b, initial probability vector x(O), normalization after each iteration step, convergence criterion

NUMERICAL SOLUTION: ITERATIVE METHODS

Fig. 3.6

A birth-death

model of two customers traversing three stations.

Tab/e 3.6 State probabilities for Fig. 3.6 computed using Grassmann’s algorithm 0.6578947368 0.1315789474 0.1315789474 0.02631578947 0.02631578947 0.02631578947

TO Tl r2 r3 x4 r5

139

Table 3.7 State probabilities for Fig. 3.6 computed using Jacobi’s method 0.6578947398 0.1315789172 0.1315789478 0.02631578619 0.02631578846 0.02631582059

TO Tl 772 R3 r4 r5

E, and the norm for test of convergence, Jacobi’s method is applied:

I I 1I 1 b = (O,O, (4%(JO>, x(o)= I6’6’6’6’6’6 xU4 x@) = IIx(lc)lll?

Ip E = lo-7,

- X(k--1q2 11x@-1) 112

> ’ < E *

The results of using Jacobi algorithm, needing 324 iterations are given in Table 3.7. The method of Jacobi is of less practical importance due to its slow pattern of convergence. But techniques have been derived to speed up its convergence, resulting in well-known algorithms such as Gauss-Seidel iteration or the successive over-relaxation (SOR) method. 3.4.4

Gauss-Seidel

Method

To improve convergence, a given method often needs to be changed only slightly. Instead of deploying full parallelism by evaluating Eq. (3.67) for each state j E S independently, we can serialize the procedure and take advantage of the

already updated new estimates in each step. Assuming the computations to be arranged in the order 0, 1, . . . , n - 1, where ISI = n, it immediately follows,

140

that

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

for the calculation

of the estimates

xj (k) , all j previously

computed

esti-

mates xik), i < j, can be used in the computation. Taking advantage of the more up-to-date information, we can significantly speed up the convergence. The resulting method is called the Gauss-Seidel iteration: b. _ 3

‘2

a.

.x(k) 23 2

+

Xi(kc) _-

ncl

a.

i=j+1

i=o

.x(k-l) 2.7 i

vj

7

E s.

(3.69)

aii

Note that the order in which the estimates xjk) are calculated in each iteration step can have a decisive impact on the speed of convergence. In particular, it is generally a matter of fact that matrices of Markov chains are sparse. The interdependencies between the equations are therefore limited to a certain degree, and parallel evaluation might still be possible, even if the most up-to-date information is incorporated in each computation step. The equations can deliberately be arranged so that the interdependencies become more or less effective for the convergence process. Equivalently, the possible degree of parallelism can be maximized or minimized by a thoughtful choice of that order. Apparently, a trade-off exists between the pattern of convergence and possible speedup due to parallelism. In matrix notation, the Gauss-Seidel iteration step is written as: xc’) =

b + x(‘-l)L

>

(D - u)-’

Equivalently, we could rewrite Eq. (3.70), thereby step from Eq. (3.69) more obviously: b + x(‘“)U

3.4.5

The Method

of Successive

+ x(‘“-l)L

>

,

k 2 1.

reflecting

D-l,

(3.70) the Gauss-Seidel

k > 1.

(3.71)

Over-Relaxation

Though Gauss-Seidel is widely applied, further variants of the iteration scheme do exist. A promising approach for an increased rate of convergence is to apply extrapolations via the so-called method of successive over-relaxation, SOR for short. The SOR method provides means to weaken or to enhance the impact of a Gauss-Seidel iteration step. A new estimate is calculated by weighting (using a relaxation parameter w) a previous estimate with the newly computed Gauss-Seidel estimate. The iteration step is presented in matrix notation in Eq. (3.72)) from which it can be concluded that the SOR method coincides with Gauss-Seidel if w is set equal to one: xc’) =

wb + x (‘--l)

(wL + (1 -w)

D))

(D - wU)-’

,

k 2 1.

(3.72)

the splitting of matrix Equation (3.72) can be easily verified. Remembering A = D - L - U and introducing a scalar w, we can immediately derive the

NUMERICAL SOLUTION: ITERATIVE METHODS

141

equation wb = xw (D - L - U) f rom Eq. (3.64) as a starting point. Adding xD to both sides of the equation yields: wb+xD=xw(D-L-U)+xD. By applying simple arithmetic

manipulations

wb+x(wL+(l-w)D)=x(D-wU),

(3.73) we get: (3.74)

and, finally, by multiplying both sides of Eq. (3.74) with (D - wU)-‘, we get Eq. (3.72). T o complete the description of the SOR algorithm, we need to focus on the impact of the numerical properties of the iterative method on the convergence process. First of all, a sensible choice of the relaxation parameter w is crucial for an accelerated convergence. Then we need to carefully specify appropriate criteria to test convergence in each iteration step. It is wise to choose these criteria as a function of the problem domain, i.e., in a model-specific manner. Finally, overflows and underflows have to be dealt with, especially if the investigated models contain states with very small probabilities associated with them. This situation commonly occurs if very large or so-called stiff models are investigated, such that some states have a very small probability mass. 3.4.5.1 The Relaxation Parameter The relaxation parameter w was introduced to gain an increased speed of convergence. It was mentioned that the incorporation of the normalization condition xl = 1 in each iteration step can have a negative impact on the rate of convergence. From this point of view, normalization should, for practical reasons, be postponed until convergence has been assured. Henceforth, we continue discussion of the iterative met hod on the basis of a homogeneous linear system 0 = xA according to Eq. (3.64). With this variant in mind, the solution of Eq. (3.72) can be identified as the largest left eigenvector yi of the matrix (wL + (1 - w) D) (D - wU)-‘. For the corresponding eigenvalue, we have IX,] = 1. (Note that the normalized eigenvector equals the desired solution vector x.) It is known from [StGo85] that the speed of convergence is a function of the second largest eigenvalue X2, with 1x11 2 IX:!), of the iteration matrix, which can be influenced by an accurate choice of w. To guarantee convergence, all eigenvalues should be real valued, the second largest eigenvalue [X2( should be smaller than one in magnitude because the smaller JX2I is, the faster the convergence would be. Conditioned on the assumption that 1x2) < 1, which is not always true, the following relation holds for the number of iteration steps i, the second largest eigenvalue (X2I, and the required accuracy E:

w4 i z log(]X2])’

(3.75)

If all the eigenvalues were known, an optimal we that minimizes IX2] of the iteration matrix could be derived. But this computation is generally

142

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

more expensive than the gain due to the implied acceleration of the convergence. Therefore, heuristics are often applied and instead of an exact value we, an approximation ~0” is derived. Sometimes advantage is taken of certain regularities of the parameter matrix for a simplified estimation wr of we. Equivalently, the parameter matrix can sometimes be rearranged in a problem-dependent way so that a more favorable structure results. For the SOR iteration method to converge, it has been shown in [HaYo81] that the relaxation parameter must obey the relation 0 < w < 2. It is usually a promising approach to choose c3 adaptively by successive reestimations of LJ:. In particular, we follow [StGo85] for such a technique. With the subsequently introduced approach, w should be chosen such that w 5 WE, otherwise convergence is not guaranteed because of possible oscillation effects. Furthermore, we assume w.M > _ 1, i.e., underrelaxation is not considered. The computation would be initiated with tic11 = 1, which is fixed for some, say 10, steps of iterations. Using the intermediate estimate ]Xr(“)I of the second largest eigenvalue, which is also called the subdominant eigenvalue, in the iteration process an intermediate estimate WE(~) of the optimal relaxation parameter can be derived by evaluating Eq. (3.76):

Wr(k) =l+&y* The estimated subdominant eigenvalue ]A?(‘) ] is approximated as a function of the recent relaxation parameter w(‘-‘) and the relative change in values S(‘-r) of the successive iterates x(‘-~), x(lcm2), x(‘-‘) as shown in Eq. (3.77):

+ (p-1) - 1 p(k) _- ($(‘“-1) 2 I I w(“-l)j/jFG Finally, the relative change in values &‘-l) diate results as presented in Eq. (3.78): SC’“-1) =

(3.77)

is calculated from the interme-

IIX(k--l) - X(k-2) 11~ llx(“-2)

- x(k--3) Iloo *

(3.78)

Note that the maximumvector norm ]]x]]~ is defined as ]]x]]~ = maxr
NUMERICAL SOLUTION: ITERATIVE METHODS

143

3.4.5.2 To terminate an iterative computation proThe Test of Convergence cess, appropriate criteria need to be provided by means of which it can be assured that the iteration process converges to the desired solution x. Generally, some vector norm 1Ix - xc’) 11would be used to estimate the error that should be smaller than some threshold E. Since x is not known, it must either be estimated as shown in [StGo85], or some different heuristics have to be applied. Intuitively, convergence is reached if successive iterates do not differ substantially. But what “substantial difference” means can only be answered in the context of the model under consideration. Convergence could be relatively slow, that is, the vectors x(‘-‘), x(‘) would not be much apart, even though still being far from the final solution x. Furthermore, if the underlying model is large, the elements xi(k) of x(‘) could easily become smaller than E and convergence could be mistakenly assumed. To overcome these two problems, it is often a good choice to use the criterion from Eq. (3.79) of the relative change in values of the most recent iterate x(‘) and the vector obtained m > 1 steps earlier, x(lcWm), namely:

IIX(k)- X(k-m)lloo< E Wk) II00 *

(3.79)

According to [StGo85], m should be chosen as a function of the number of necessary iterations, 5 < m < 50. If SOR converges quickly, the much simpler and computationally more efficient criterion as given in Eq. (3.80) might be applicable:

11x(“)- X(k-l))) < E.

(3.80)

Commonly applied vector norms are:

lb41= 2 14, i=l

Il4I2

=

g- Ix?. \

(3.82)

i=l

3.4.5.3 Overflow and Underflow If the normalization condition is incorporated in the transition matrix, as discussed in the context of Eq. (3.64), overflow and underflow are less likely to occur, but convergence is much slower and computational overhead is higher. Therefore, some heuristics are applied to circumvent these numerical problems. To avoid underflow, elements xi(k) shrinking below a lower threshold el are simply set equal to zero. Conversely, if there exist elements xi(ICIthat exceed an upper threshold Ed, a normalization is adaptively performed and the values are scaled back to a more convenient range.

144

STEADY-STATE SOLUTIONS OF MARKOV CHAlNS

3.4.5.4 The Algorithm The algorithm presented in Fig. 3.7 is in a skeletal form. As we have discussed at length, it needs to be carefully adapted for the peculiarities of specific cases to which it is applied. Since for w = 1 SOR coincides with Gauss-Seidel, we do not separately present the Gauss-Seidel algorithm. Furthermore, for the sake of simplicity, we do not consider scaling and resealing of the parameter matrix, which is a technique that is commonly performed in order to avoid divisions by diagonal elements ajj (cf. Eq. (3.69)). It should be noted that a sum indexed from 0 to -1 is interpreted to be zero,

3.5

COMPARISON

OF NUMERICAL

SOLUTION

METHODS

The solution algorithms discussed in earlier sections are herein compared by the means of the model introduced in Fig. 2.7 (Section 2.2.2). Patterns of convergence and numerical accuracy are compared. According to Eq. (2.58), the following system of steady-state equations results: 0 = -2yn2 + 67r1, 0 = -prRC + 2c77@, 0 = -OTRB + 2(1- C)y;lrz, 0 = -(Y+S)q 0 =

-&p-J

+

QKRB

+ @RC

+ SrO,

+ y7r1.

With the normalization condition:

and the parameters from Table 3.8, three computational experiments were performed using the Gaussian elimination algorithm. The results are summaTable3.8 Experimental parameters for the model with two processorelements Parameter

Experiment 24 hr 54 set 4 hr 10 min 10 min 30 set 0.99

1

Experiment 1 yr 12 hr 4 hr 10 min 10 min 30 set 0.99

2

Experiment

3

100 yr 262 hr 48 min 4 hr 10 min ’ 10 min 30 set 0.99

rized in Table 3.9. From the data of the first experiment it is concluded that, for example, with probability 7r2+ ~1 w .71+ .25 = .96, at least one processor is up in steady state. This quantity could provide an important measure for

the availability of a modeledsystem. Other measuresof interest can be similarly derived. By comparing the results of different experiments in Table 3.9

COMPARISON OF NUMERICAL SOLUTION METHODS

I-

84lve

xl=

1 Fig. 3.7

The SOR iteration algorithm.

145

146

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

Table 3.9

Steady-state

Probability r2 TRB TRC

m TO

probabilities Experiment

1

0.7100758490525 0.0000986153339 0.0004881459029 0.2465383347910 0.0427990549197

of the model with two processor elements Experiment

2

0.9990481945467 0.0000003796383 0.0000018792097 0.0009490957848 0.0000004508205

Experiment

3

0.9999904674119 0.0000000038040 0.0000000188296 0.0000095099093 0.0000000000452

it can be seen how the state probabilities depend on the MTTF, being varied from approximately 1 day to a little less than 100 years and 11 days. Although the probability mass is captured to a larger extent by ~2, the other probabilities are of no less interest to designers of dependable systems because the effectiveness of redundancy and recovery techniques is measured by the values of these small probabilities. Note, in the case of the third experiment in particular, that the numbers differ by many orders of magnitude. This property could impose challenges on the numerical algorithms used to analyze the underlying CTMC since an appropriate accuracy has to be assured. For instance, the results in Table 3.9 are exact values up to 13 digits. While such an accuracy is most likely not very significant for the probability ~2, it seems to be a minimum requirement for the computation of OTTO, since the 10 leading digits of ~0 are zeroes in the third experiment. The more the values of the model parameters diverge from each other and the larger the investigated model is, the more this property tends to manifest itself. Note that for the analysis of larger models, iterative methods are usually preferred. The convergence criteria have to be chosen very carefully in cases where the resulting values differ by orders of magnitude. On the one hand, the convergence criteria should be stringent if the significant measures result in very small values. One the other hand, computational complexity increases substantially with a more stringent convergence criteria. Even worse, the probability of convergence might significantly decrease. More details are presented in the following sections. 3.5.1

Case Studies

In the following, we refer back to the model in Fig. 2.7 as a starting point. The model can easily be extended to the case of n processors, while the basic structure is preserved. Furthermore, the same parameters as in Table 3.8 are used in three corresponding experiments. The computations are performed with Gauss elimination, the power method, the Gauss-Seidel algorithm, and different variants of the SOR method. The intention is to study the accuracy and pattern of convergence of the iterative methods in a comparative manner. Concerning the SOR method, for example, no sufficiently efficient algorithm

COMPARISON OF NUMERICAL SOLUTION METHODS

147

is known for a computation of the optimum w. With the help of these case studies some useful insights can be obtained. There are many possibilities to specify convergence criteria, as has been discussed earlier. In our example, convergence is assumed if both a maximum norm variant: (3.83) of the difference between the most recent iterates xcn) and x(+l) square root norm:

and the

In-1

lb% =

C(d2, 11i=o

(3.84)

of the residues: i-l ri

= C

n-l qj,ixy)

j=o

+ C

qj,ixi”-”

-

bi,

(3.85)

j=i

are less than the specified E:

Note that the pattern of convergence of iterative methods can depend on the ordering of the states as well, and thus on the iteration matrix. This dependence is particularly important for the SOR-type methods where in each iteration step the most up-to-date information is used in the computation of the components of the intermediate solution vector. Therefore, we provide in our comparative study the results of two different state ordering strategies. The resulting number of iterations are summarized in Table 3.10 for the cases of E = 10-l’, 10-15, 10-l”, 10-17, and 10m20as some examples. Comparing corresponding experiments, it can be seen from Table 3.10 how the number of iteration steps of the power method increases as the convergence criterion is tightened from E = 10-l’ to E = 10e2’. Furthermore, the power method tends to converge faster, the more the model parameters differ quantitatively from each other. In these examples, convergence is relatively slow in experiment one, medium in experiment two, and fast in the third experiment for fixed E. A similar pattern of convergence using the Gauss-Seidel method and the SOR variants can be observed. The relative best results are achieved by SOR with w = 0.8. Gauss-Seidel and SOR (w = 1.2) do not converge if Another view is provided in Fig. 3.8 where the number of iterations is depicted as a function of w that is systematically varied, 0 < w < 2, and the criterion of convergence E = 10-15 is chosen. Entries on the graph indicate that convergence was observed within 20,000 iterations for a particular w.

148

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

Table 3.10

Number of iterations required to analyze the model with ten processors e= 1.e-10

e= l.e-15

Methods

Exp. 1

Exp. 2

Exp. 3

Exp. 1

Exp. 2

Exp. 3

Power Gauss-Seidel SOR (w = 1.2) SOR (w = 0.8)

32119 653

16803 23 49 41

16703 17 41

61013 *

24305 25 55 47

24083 17 41

2;9

3:7

e= lee-l6

e = 1. e-l7

Methods

Exp. 1

Exp. 2

Exp. 3

Exp. 1

Exp. 2

Exp. 3

Power Gauss-Seidel SOR (w = 1.2) SOR (w = 0.8)

66547 * * 369

26487 25 57 47

25309 17

70221 * * *

26883 *

26625 *

4;

4:

4:

f~1.e~~~ Methods

Exp. 1

Exp. 2

Exp. 3

Power Gauss-Seidel SOR (w = 1.2) SOR (w = 0.8)

71573 * * *

30039 * * *

30703 * 4*7

Note: An asterisk (*) indicates no convergence in 20,000 iterations

and a dash (-) indicates

convergence with negative results.

6000

4000

Iterations 3000

exp. 1

exp. 2 ,___r_____--

012

Fig. 3.8

014

0.6

0:s

i

"'

1:2

-e-e-

""

The number of SOR iterations as a function of w.

COMPARISON OF NUMERICAL SOLUTION METHODS

149

The results confirm that convergence is relatively slow in experiment one. Furthermore, in experiment one, SOR converges only if w < 1 while the number of iterations are quite sensitive to variations in ‘w. Fewer iteration steps are needed in experiments two and three. Relatively fewer iterations, 279, are observed in experiment one with w = 0.95, in experiment two, 25, with w = 1.0; and in experiment three, 21, with w = 1.01. It is interesting to note that in experiments one and three, even if w is chosen only slightly larger than 0.95 and 1.01, respectively, no convergence results. Similar outcomes are observed for different criteria of convergence E. Over-relaxation, that is w > 1, can only be successfully applied in experiment two.

Table3.11 Accuracy (A) in the number of digits and number of iterations (I) needed by different SOR variants with E = lo-l5 Exp. 2 (d)

Exp. 2 (r)

Exp. 3 (d)

Exp. 3 (r)

W

A

I

A

I

A

I

A

I

0.1 0.2 0.5 0.7 0.8 1.0 1.1 1.2 1.3

* 21 22 23 23 33 28 23 23

* 279 97 59 47 27 41 57 81

20 20 20 22 23 23 23 23 23

539 273 91 57 47 25 39 55 79

2: 23 23 24 49 45 * *

2:5 91 55 43 21 45 * *

20 20 21 21 23 30 -

507 247 83 51 41 17 -

Note: An asterisk (*) indicates no convergence in 20,000 iterations

and a dash (-) indicates

convergence with negative results.

Using different norms affects pattern of convergence and accuracy of the numerical results even for the same criterion of convergence E. In Fig. 3.9 the number of iterations is depicted as a function of w in experiment three, comparing the residue norm (T) to the difference norm (d) with E = 10-15. It can be seen that applying the difference norm consistently results in slightly more iterations for a given w. A similar pattern was observed in all the experiments. There are ranges of w where the application of the difference norm does not lead to convergence while the application of the residue norm does lead to convergence, and vice versa. The accuracy in terms of the number of correct digits of the numerical values resulting from the application of different norms in the SOR algorithm is indicated in Table 3.11. The numbers are compared to the results obtained using Gauss elimination. It can be seen that the difference norm, both in experiments two and three, consistently results in the same or higher accuracy as the residue norm for a given w. A trade-off is observed between the number of necessary iterations and the achieved accuracy. Both factors have to be taken into account when designing calculations.

150

STEADY-STATE SOLUTIONS OF MARKOV CHAINS

In experiment four, the data from Table 3.12 are applied computations. The number of iterations versus w is depicted The sensitivity of the convergence behavior to w is impressive While the minimum number of iterations are observed with number of iterations grows quickly as w approaches one.

in a series of in Fig. 3.10. in this study. w NN0.8, the

:, exp. 3(r) ,\ I ,

0

l.:::l::,:,,::,l.:::I:

::I::.:

0:2

0.4

0:6

0:8

W

Fig. 3.9

Comparison

of difference

(d) and residue

(T) norm in experiment

Table 3.12 Experimental parameters the model with ten processors Parameter

Experiment

three.

for

4

9512 yr 8243 hr 24 min

50 min 5 min 30 set 0.99

norm

10-20 residue

We do not claim to be complete in the discussion of factors that have an impact on the speed of convergence of numerical techniques. We only presented some issues of practical relevance that have to be taken into account while considering convergence. The results presented are conditioned on the fact that other convergence dominating issues such as state ordering were not

COMPARISON OF NUMERICAL SOLUTION METHODS

151

800--

600~-

Iterations 400~-

200~-

01

0:2

0.4

0.6

0.8

1

W

Fig. 3.10 Number of iterations applying the square root norm llrjlz < 10m2’ of the residue in experiment four.

taken into account. The methods could have been tuned further, of course, resulting in a different pattern of convergence.

Steady-State Aggregation/ D&aggregation Methods In this chapter we consider two main approximation methods: Courtois’s decomposition method and Takahashi’s iterative aggregation/disaggregation method.

4.1

COURTOIS’S

APPROXIMATE

METHOD

In this section we introduce an efficient method for the steady-state analysis of Markov chains. Whereas direct and iterative techniques can be used for the exact analysis of Markov chains as previously discussed, the method computations of Courtois [Cour75, Cour77] is mainly applied to approximate u NNof the desired state probability vector u. Courtois’s approach is based on decomposability properties of the models under consideration. Initially, substructures are identified that can separately be analyzed. Then, an aggregation procedure is performed that usesindependently computed subresults as constituent parts for composing the final results. The applicability of the method needsto be verified in each case. If the Markov chain has tightly coupled subsetsof states, where the states within each subset are tightly coupled to each other and weakly coupled to states outside the subset, it provides a strong intuitive indication of the applicability of the approach. Such a subset of states might then be aggregated to form a macro state as a basis for further analysis. The macro state probabilities, together with the conditional micro state probabilities from within the subsets, can be composed to yield the micro state probabilities of the initial model. Details of the approach are clarified through the following example. 153

154 4.1.1

STEADY-STATE AGGREGATION/ DLSAGGREGATION METHODS

Decomposition

Since the method of Courtois is usually expressed in terms of a DTMC, whereas we are emphasizing the use of a CTMC in our discussion of methodologies, we would like to take advantage of this example and bridge the gap by choosing a CTMC as a starting point for our analysis. With the following model in mind, we can explain the CTMC depicted in Fig. 4.1. We assume a system in which two customers are circulating among three stations according to some stochastic regularities. Each arbitrary pattern of distribution of the customers among the stations is represented by a state. In general, if there are N such stations over which K customers are arbitrarily distributed, then from simple combinatorial reasoning we know that (“$“F’) = (“‘,“-‘) such combinations exist. Hence, in our example with N = 3 and K = 2 we have 4 02 = 6 states.

Fig. 4.1

CTMC subject to decomposition.

In state 020, for example, two customers are in station two, while stations one and three are both empty. After a time period of exponentially distributed length, a customer travels from station two to station one or station three. The transition behavior is governed by the transition rates ,921or ~23 to states 110 or 011, respectively. The transition behavior between the other states can be explained similarly. The analysis of such a simple model could be easily carried out by using one of the standard direct or iterative methods. Indeed, we use an exact method to validate the accuracy of the decomposition/aggregation approach for our example. We use the simple example to illustrate Courtois’s method. To this end, the model needs to be explored further as to whether it is nearly completely decomposable, that is, whether we can find state subsets that represent tightly coupled structures. The application may suggest a state set partitioning along the lines of the customers’ circulation pat tern among the visited stations. This would be a promising approach if the customers are preferably staying within the

COURTOIS’S APPROXIMATE

’ , : ’ ‘\ :

METHOD

155

/’ /’ ,’ , 8 8’ ,’

1‘I,g 0 8002

Fig. 4.2 Decomposition of the CTMC with regard to station three.

Fig. 4.3

Decompositions

of the CTMC with regard to stations two and one.

bounds of a subset of two stations and only relatively rarely transfer to the third station, i.e., the most isolated one. All such possibilities are depicted in Fig. 4.2 and 4.3. In Fig. 4.2, station three is assumed to be the most isolated one, i.e., the one with the least interactions with the others. Solid arcs emphasize the tightly coupled states, whereas dotted arcs are used to represent the “loose coupling.” When customers are in the first station, they are much more likely to transit to the second station, then return to the first station, before visiting the third station. Hence, we would come up with three subsets {020,110,200}, {011, lOl}, and {002} in each of which the number of customers in the third station is fixed at 0, 1, and 2, respectively. Alternatively, in Fig. 4.3, we have shown the scenario where stations two and three are isolated. Now we proceed to discuss Courtois’s method on the basis of the set of parameters in Table 4.1. It suggests a decomposition according to Fig. 4.2. Clearly, the parameter values indicate strong interactions between stations one and two, whereas the third station seems to interact somewhat less with

156

STEADY-STATE

AGGREGATION/

DISAGGREGATION

Tab/e

Transition

4.1

rates = 0.20 jai32 = 0.20

1-121= 2.40 /A.23= 0.15

p12 = 4.50 p13 = 0.30

METHODS

p31

This level of interaction should be reflected at the CTMC level representation. The corresponding infinitesimal generator matrix Q is given

the others. ES:

-4.80 2.40

Q=

0 0.20

0 0

0

4.50 -7.35 2.40 0.20 0.20

0

0.30 0.15

4.50 -2.55

0.30 0.15 4.50

0

0 -5.20 0.20

0

2.40 0.20

0

-2.95 0.20

0 0 0 0.30 0.15 -0.40

and is depicted symbolically in Table 4.2.

Tab/e 4.2 Generator matrix structured for decomposition

200

110

200

-c

l-412

110

P21

-c

P12

020

0

1121

-c

101

P31

P32

0

011

0

P31

002

0

0

I

020 0

I

Q of the CTMC 101 P13

CL23 0

011

002

0 Pl3

P.23

-c

i-412

P32

P21

-c

0

P31

l-W.2

W3

It is known from Eq. (3.5) that we can transform any CTMC to a DTMC by defining P = Q/q + I, with Q > maxi,jES I~ijl. Next we solve v = VP instead of 7rQ = 0 and we can assert that v = 7r. Of course, we have to fulfill the normalization condition vl = 1. For our example, the transformation results in the transition probability matrix as shown in Table 4.3, where q is appropriately chosen. Given the condition g > max;,j )g;jJ = 7.35, we conveniently fix 4 = 10. Substituting the parameter values from Table 4.1 in Table 4.3 yields the tran-

COURTOIS’S

Tab/e 4.3 Transition-probability structured for decomposition

200 200 110 020

l-5

l-5

011

002

0

Lui4

0

0

y

1-F

so11

fz!a2 Q .&u 4

?!siz 4

002

0

0

0

0

0 1-F &u4 l!A3.l 4

157

P of the DTMC

101

y

&LL 4

METHOD

020

y

l!Lzl 4 0

matrix

l!Ql 4 0

101

sition

110

APPROXIMATE

y

0

l!Lzi Q

0

y 1-F t&XL 4

l!aa Q y 1-F

probability matrix P:

P=

0.52 0.45 0 0.03 0 0 0.24 0.265 0.45 0.015 0.03 0 0 0.24 0.745 0 0.02 0.02 0 0 0.02 0.02 0.24 0.705 0.015 0 0 0

(4.1)

As indicated, P is partitioned into I&! x M) number of submatrices PIJ, with A4 = 3 and 0 5 I, J 5 2 in our example. The submatrix with macro row index I and column index J is denoted by PIJ, while the elements of each submatrix can be addressed via double subscription PI Jij. The M diagonal submatrices PII represent for I = 0, 1,2 the interesting casesof zero, one, and two customers, respectively, staying in the third station. For example, the second diagonal submatrix Prr is:

The elements of the submatrix can be addressedby the schemajust introduced: pll 1,-,= 0.24. Since the indices I, 0 5 I 5 M - 1 are used to uniquely refer to subsetsas elementsof a partition of the state spaceS, we conveniently denote the corresponding subset of states by SI; for example, Sr = { 101,011). Of course, each diagonal submatrix could possibly be further partitioned into substructures according to this schema. The depth to which such multilayer decomposition/aggregation technique is carried out depends on the number of stations N and on the degree of coupling between the stations. We further elaborate on criteria for the decomposability in the following.

158

STEADY-STATE

AGGREGATION/DlSAGGREGATlON

METHODS

A convenient way to structure the transition probability example is to number the states using radix K notation: N c i=l

matrix

in our

N

kiKi,

where

c

ki = K.

i=l

where /Q denotes the number of customers in station i and N is the index of the least coupled station, i.e., the one to be isolated. For instance, in our example a value of 4 would be assigned to state 200 and a value of 12 to state 011. Note that the states in Table 4.2 and 4.3 have been numbered according to this rule. The transition probability matrix in block form is given as: /’

P=

PO0

PO1

PlO

Pll

...

. . .

PI0

PO(M-1)

. . .

PIJ

Pl(M-1)

* **

PI(M-1)

*

(4.3)

.

.

. . *

P(M-2)O

. . *

\ ,P(M-1)O

P(M-P)(M-1) q&f-1)(&l)

Generally, the partitioning of matrix P in submatrices PIJ is done as a function of the number of customers in the least, coupled station N. Given that K customers are in the system, there will be M = K + 1 diagonal submatrices. Each diagonal submatrix PII, 0 5 I 5 K, is then considered to be a basic building block for defining transition probability matrices P;I of a reduced system of N-l stations and K-I customers. For example, PT, would be used to describe a system with K - 1 customers circulating among N - 1 stations, while one customer stays in station N all the time. The transitions from station N to all other stations would be temporarily ign0red.l In terms of our example, we would only consider transitions between states 101 and 011. To derive stochastic submatrices, the transition probability matrix P is decomposed into two matrices A and B that add up to the original one. Matrix A comprises the diagonal submatrices PII, and matrix B the complementary off-diagonal submatrices PIJ, I # J: P=A+B.

(4.4)

In our example, Eq. (4.4) has the following form: Ot ‘Note that P11 is not a stochastic matrix; matrix

PT1.

it has to be modified to obtain a stochastic

COURTOIS’S

P=

/0.52 0.45 0 0.24 0.265 0.45 0 0.24 0.745

0

\

0.96

METHOD

0.03 0 0.015 0.03 0 0.015

0 0 0 0.02 0.02 0 0.03 0 0.02 0.02 0 0.015 0 0 0 0.02 0.02 0 0

0.48 0.45 0.24 0.705

0

APPROXIMATE

A

B

159

I

Since we wish to transform each diagonal submatrix PII into a stochastic matrix PT,, we need to define a matrix X, such that Eq. (4.5) holds and matrices P*, P&, . . . PT,-,)(,-,) are all stochastic:

P*=(A+X)=

There are multiple ways to define matrix X. Usually, accuracy and comput ational complexity of the method depend on the way X is defined. It is most important, though, to ensure that the matrices P;,,O 5 I 2 M - 1, are all ergodic, i.e., they are aperiodic and irreducible. Incorporating matrix X in matrix P, we get: P=(A+X)+(B-X) (4.6)

=p*+c.

In our example, we suggest the use of following matrix X in order to create a stochastic matrix P**. 0

0

0.03

0

0

0 \

x=

This gives us:

P=

'0.52 0.45 0.03 0.24 0.31 0.45 0.015 0.24 0.745 0

C

160

STEADY-STATE

4.1.2

Applicability

AGGREGATION/

DISAGGREGATION

METHODS

The applicability of Courtois’s method needs to be checked in each case. In general, partitioning of the state space and subsequent aggregation of the resulting subsets into macro states can be exactly performed if the DTMC under consideration has a lumpable transition probability matrix. In such a case, the application of Courtois’s method will not be an approximation. A transition probability matrix P = bij] is lumpable with respect to a partition of S in subsets SI, 0 5 1 5 n/r - 1, if for each submatrix PIJ,V’I, J,O 5 I # J 5 111- 1 real-valued numbers 0 < ~IJ 5 1 exist such that Eq. (4.7) holds [KeSn78]:

c

=

PIJij

vi

“+IJ,

E SI.

(4*7)

jESJ

Note that the diagonal submatrices PII need not be completely decoupled from the rest of the system, but rather the matrix has to exhibit regularities imposed by the lumpability condition in order to allow an exact aggregation. In fact, if the PII are completely decoupled from the rest of the system, i.e., the r~J = 0, I # J, then this can be regarded as a special caseof lumpability of P. More details on and an application example of state lumping techniques are given in Section 4.2, particularly in Section 4.2.2. From our example in Eq. (4.1), P is not lumpable with respect to the chosen partition. Hence Courtois’s method will be an approximation. A measureof accuracy can be derived according to Courtois from Eqs. (4.4) and (4.6). The degree Eof coupling between macro states can be computed from matrix B = [bij] in Eq. (4.4). If E is sufficiently small it can be shown that the error induced by Courtois’s method is bounded by O(E). Let Ebe defined as follows:

(44

e=max iES

Eq. (4.6) can be rewritten as: P=P*+&.

(4-g)

In our example, Ecan be computed to be: 0.03 0.015 0.03 0.015 0.02 0.02 0.03

+

E=max i

0.02

+ + 1=0-07* t 0.02 t 0.015 0.02 +0.02

COURTOIS’S

APPROXIMATE

161

METHOD

To prove that P is nearly completely decomposable, it is sufficient to show that Relation (4.10) holds between E and the maximum of the second largest eigenvalues A;(2) of P;, for all I, 0 5 I 5 M - 1 [Cour77]:

E < 1 - maw IAl;@) 2

(4.10)

-

For each PT,, the eigenvalues A; (k) can be arranged according to their decreasing absolute values: (4.11)

IW)I > IJw>l > *-* > Im-wl~

Since P;,, 0 5 I 5 2, are all stochastic, we can immediately conclude Ix;(l)( = 1 for all I. In our example, the eigenvalues of the three diagonal submatrices:

need to be computed. equation: det(P&

The eigenvalues of P&, are the roots of the following

- X:1) =

(0.52 - A;) 0.24 0.015

0.45 (0.31 - Xi) 0.24

= (XT - 1)(x; - 0.595)(x; = 0,

+ 0.02)

The resulting where I denotes the identity matrix. arranged according to their absolute value as: Ix;(l)j

= 1,

Ix;(a)/

= 0.595,

The eigenvalues of PT 1 are determined det(PT, - XTI) =

eigenvalues of P&

IX;;(3)1 = 0.02.

by the solution of:

(0.48 - A;) 0.295

0.52 (0.705 - A;)

= (A; - 1)(X; - 0.185) = 0,

0.03 0.45 (0.745 - A;)

are

162

STEADY-STATE

AGGREGATiON/DlSAGGREGATlON

METHODS

which, in turn, gives: IXi(l)I

= 1 and

IXi(2)l = 0.185.

The eigenvalue of P&, immediately evaluates to: Ix;(l)1 = 1. With the second largest eigenvalues computed, we can examine whether decomposability Condition (4.10) holds: m;xjJI;(2)/

= max

L

od5ps8\ = 0.595 1

0.07 = E< 1 - maxl IX”;(Z)1 2 1 - 0.595 2 = 0.2025.

o.o7=e<

Because E < 0.2025, Condition (4.10) holds and the transition probability matrix P is nearly completely decomposable with respect to the chosen decomposition strategy depicted in Fig. 4.2. 4.1.3

Analysis

of the Substructures

As a first step toward the computation of the approximate state probability vector v” w u, we analyze each submatrix P;I separately and compute the conditional state probability vector Y;, 0 5 I 5 M - 1: qp;,

up=

- I) = 0,

1.

(4.12)

Thus V; is the left eigenvector of Pf, corresponding to the eigenvalue X;(l) = 1. Substituting the parameters from P& of our example, the solution of

(yg*cvvo*P42) -

(0.52 - 1) 0.24 0.015

0.45 (0.31 - 1) 0.24

0.03 0.45 (0.745 - 1)

= 0,

YZl = 1

yields the conditional steady-state probabilities of the micro states aggregated to the corresponding macro state 0: z&, = 0.164,

u& = 0.295,

l& = 0.54.

Similarly, PT, is used in

(

40’4,)

-

(0.48 - 1) 0.295

(

0.52

(0.705 - 1)>

=o,

VT1 = 1

COURTOIS’S

APPROXIMATE

METHOD

163

to obtain the conditional state probabilities of the micro states aggregated to the corresponding macro state 1: l&

= 0.362,

z& = 0.638.

Finally:

Nearly completely decomposablesystems can be characterized with respect to “long-term” and “short-term” behavior: l

From a “short-term” perspective, systems described by their probability transition matrix P can be decomposedinto M independent subsystems, each of whose dynamics is governed by a stochastic processapproximately described by the matrices PF, , O
l

In the “long run,” the impact of the interdependencies between the subsystems cannot be neglected. The interdependencies between subsystems, or macro states, I and J, for all I, J, are described by the transition probabilities IIJ that can be - approximately - derived from the transition probability matrix P and the conditional state probability vector UT. Solving the transition matrix P = [I’,J] for the macro states yields the macro-state probability vector y. The macro-state probability vector y can, in turn, be used for unconditioning of UT, 0 5 I 5 Ad- 1, to yield the final result, the approximate state probability vector V” * V.

4.1.4

Aggregation

and Unconditioning

Having obtained the steady-state probability vector for each subset, we are now ready for the next step in Courtois’s method. The transition probability matrix over the macro states, T’ = [I’ 1~] is approximately computed as: (4.13)

Comparing Eq. (4.13) with the lumpability Condition (4.7), it is clear that the I’IJ can be exactly determined if the model under consideration is lumpable, that is: hJ

holds, independent of v;.

= rIJT

(4.14)

164

STEADY-STATE

AGGREGATION/

DISAGGREGATION

In our example, the macro-state according to Eq. (4.13):

METHODS

transition

probabilities

I~J are derived

roe= 40.(PO000 +PO001 +poo()J +z&.(pool0 +pool1 +POO12) +42.(PO020 +PO021+ poo22) = 0.9737, l-b1

=

uto* +

(PO100 42.

r02

=

40.

(P0200)

ho

=

40.

ho00

‘ll

=

‘TO*

hloo

I721

h3

+

+

vile

(poll0

(p02~~) +

+ Pllol)

- rll

V&b

+ POlll)

= 0.0263,

+ ~01~~)

+ moo1

r12 = 1 -ho r20

+Po101)

(Po120

+

+

uT3-

PlOo2)

+

&*

(JIlllo

(pozzo)

z&.

=

(plolo

1 -

roe

+ PIOll

+ pllll)

=

_ rol

=

+ plo12)

0, =

0.04,

0.9396,

= 0.0204,

=

40’

(P2000

+ p2001

=

G

(Pa100

+ p2101)

+ p2002) =

=

0,

0.04,

= 1- r20- r21= 0.96.

Accordingly, the macro-state transition probability matrix r is:

I?=

0.9737 0.04 0

0.0263 0.9396 0.04

0 0.0204 0.96

The next step is to compute the macro steady-state probability vector y using:

Yr=Y,

yl=

1.

(4.15)

In our example, Eq. (4.15) results in: (0.9737 - 1)

0.0263

0.04

(0.9396 - 1)

0

0.04

(YO/YlrY2)

0.0:04 (0.96 - 1)

= 0,

yl = 1,

from which the macro steady-state probabilities are calculated as:

YO = 0.502,

yl = 0.330,

y2 = 0.168.

The final step is to obtain the approximate steady-state probabilities for the original model. For convenience, the elements of the state probability vector vz E u are partitioned along the lines of the decomposition strategy. Hence, we refer to the elements of V” = [v?J in compliance with the usual notation in this context. The unconditioning in Eq. (4.16) of the VT is to be performed for all macro states I: qi

=

YI’TiT

0 5 I 5 M - 1 and V’i E SI

(4.16)

COURTOIS’S

Tab/e 4.4

State probabilities

State

Probability

200 110 020 101 011 002

6J Gl G2 +I U1”l UF

computed Formula Yo& You;1 70 u; 2 Yl$o

Yl& Y2$

APPROXIMATE

using Courtois’s Approx. 0.0828 0.1480 0.2710 0.1194 0.2108 0.1680

165

METHOD

method and exact ones Exact 0.0787 0.1478 0.2777 0.1144 0.2150 0.1664

% Error + + + +

5.2 0.1 2.4 4.4 2.0 0.1

The use of Eq. (4.16) in our example gives the results summarized in Table 4.4. For the sake of completeness, the exact state probabilities are also shown for comparison. For models with large state spaces, Courtois’s method can be very efficient if the underlying model is nearly completely decomposable. If the transition probability matrix obeys certain regularity conditions the results can be exact. The error induced by the method can, in principle, be bounded [CoSe84]. But since the computational complexity of this operation is considerable, a formal error bounding is often omitted. The efficiency of Courtois’s method is due to the fact that instead of solving one linear system of equations of the size of the state space S, several much smaller linear systems are solved independently, one system for each subset SJ of the partitioned state space S, and one for the aggregated chain. We conclude this section by summarizing the entire algorithm. For the sake of simplicity, only one level of decomposition is considered. Of course, the method can be iteratively applied on each diagonal submatrix P;,. 4.1.5

The

Algorithm

Create the state space and organize it appropriately according to of decomposition. Build the transition probability matrix P (by use of randomization P = Q/q+1 if the starting point is a CTMC), and partition P into AL! x M number of submatrices PIJ, 0 2 I, J, 5 M - 1, appropriately. Verify the nearly complete decomposability of P according to Relawith the chosen value of E. Decompose P such that P = P* + EC according to Eq. (4.9). Matrix P* contains only stochastic diagonal submatrices P;,, and E is a measure of the accuracy of Courtois’s method. It is defined as the maximum sum of the entries of the non-diagonal submatrices PIJ, I # J, of P. For each I, 0 5 I 5 M - 1, solve equation $PT, vT1 = 1 to obtain the conditional state probability vectors UT.

= UT with

166

STEADY-STATE AGGREGATION/ DISAGGREGATION METHODS

Compute the coupling between the decomposed macro states: -1 Generate the transition to Eq. (4.13). -1 vector y.

probability

matrix I’ = [l?~,] according

Solve Eq. (4.15) to obtain the macro steady-state probability

Compute the approximate steady-state probability vector zP of the micro states by unconditioning of the conditional state probability vectors vT,O 5 I < &! - 1, according to Eq. (4.16). From v” compute steady-state performance and dependability measures along the lines of a specified reward structure. Problem 4.1 Verify that the transition probability is not lumpable according to the chosen partition.

matrix from Eq. (4.1)

Problem 4.2 Recall Eq. (4.7) and modify the parameters in Table 4.1 of the example in Fig. 4.1 such that the resulting model is lumpable and nearly completely decomposable. Derive the generator matrix according to Table 4.2 and the transition probability matrix according to Table 4.3 for the resulting model. Problem 4.3 Discuss the relationship between lumpability and nearly complete decomposability. Under what condition does Courtois’s method yield the exact state probabilities if the model is both lumpable and nearly completely decomposable ? As a hint, compare Eq. (4.13) with lumpability Condition (4.7) and relate it to Eq. (4.16). Problem 4.4 Modify Courtois’s method such that it becomes directly applicable for an analysis of CTMCs. Apply the new algorithm directly, that is, without using uniformization, to the original model in Fig. 4.1 and solve for the steady-state probability vector.

4.2

TAKAHASHI’S

ITERATIVE

METHOD

Although closely related to and under some circumstances even coinciding with Courtois’ approach, Takahashi’s method [Taka75] differs substantially both with respect to the methodology used and the applicability conditions. While Courtois’s method is non-iterative and is applied for the approximate computation of the steady-state probability vector v” for a given ergodic DTMC (or 7rTxin the case of a CTMC), Takahashi’s iterative method allows a computation of the exact state probability vector. (Note that numerical errors are still encountered even in such “exact” methods. The method is exact in that there are no modeling approximations.) To allow a straightforward comparison of the two methods, we prefer a continued discussion in terms of

TAKAHASHI’S

ITERATIVE

METHOD

167

DTMCs. Recall that any given ergodic CTMC can easily be transformed into an ergodic DTMC. Much like Courtois’s method, Takahashi’s approach is to partition the state space S into M disjoint subsets of states Sr c S, 0 5 I 5 iVl - 1 such that state I. The criteria, however, each subset Sr is aggregated into a macro used to cluster states differ in the two approaches. The calculation of transition probabilities IIJ among the macro states I and J is performed on the basis of conditional micro-state probability vector V; and the originally given transition probabilities pij, or PI Jij according to Eq. (4.13). With Courtois’s method, the conditional probability vectors UT, given partition element I, can be separately computed for each subset of micro states SI, 0 5 I 5 M - 1, if the original model is nearly completely decomposable. By contrast, in Takahashi’s method, the partitioning of the state space is performed on the basis of approximate lumpabiEity.2 Note that if the state space is approximately lumpable with respect to a partition, it does not necessarily imply that the subsets are nearly decoupled as needed in Courtois’s approach. As an immediate consequence, the conditional micro-state probabilities cannot be calculated independently for each subset of states. Instead, the complementary subset of states and the interactions with this complement is taken into account when approximating conditional micro-state probabilities of a partition element. The whole complementary set is aggregated into a single state representing all external interactions from the states within the particular subset. Since the micro-state probabilities as well as the macro-state probabilities are not known in the beginning, initial estimates are needed. The macro- and micro-state probabilities are iteratively calculated with Takahashi’s method. Aggregation and disaggregation phases are repeated alternately until some convergence criterion is satisfied. By adding an extra computational step, Schweitzer [Schw84] shows that geometric convergence can be guaranteed. Usually, the extra computation takes the form of a power or a Gauss-Seidel iteration step. 4.2.1

The

Fundamental

Equations

Our discussion of Takahashi’s method is based on that given by Schweitzer [Schw84]. Further details can also be found in [Stew94]. Let a given state space S be partitioned into M subsets: M-l s=

u

SI

and

&nsJ=&

VI#J,

O
(4.17)

I=0

2While lumpability has been later in Section 4.2.2.

introduced

in Eq.

(4.7),

approximate

lumpability

is defined

168

STEADY-STATE AGGREGATION/ DISAGGREGATION METHODS

Formally, the macro-state probability vector y could be calculated from the micro-state probability vector v for each component I of the vector: O
YI= -&

(4.18)

iESI

Since u is not known in advance, Eq. (4.18) cannot be directly used. The macro-state probabilities can be derived, however, if the macro-state transition probability matrix I? = [I’IJ] were known via the solution of Eq. (4.19): y = yr,

yl = 1.

(4.19)

The implied aggregation step is based on iterative approximations of the probabilities uTi of states i E SI given an estimate vx of the micro-state probability vector: (4.20) Therefore, from Eq. (4.20), and Eq. (4.13), th e macro-state transition probabilities ~IJ can be approximated as:

(4.21)

For the disaggregation step, the probabilities rli of transitions from macro state I to micro state i are pairwise needed for all 0 5 I 5 M - 1 and for all i E S. The transition probabilities rli can be derived from Eq. (4.21) and are given by Eq. (4.22): .

(4.22)

To complete the disaggregation step, the micro-state probabilities vi are expressed in terms of macro-state probabilities ye and transition probabilities l?r; for each subset of states Sr separately. To accomplish this task, [SI] linear equations need to be solved for each subset SI : M-l vi =

c jESI

UjPji +

c

YKrKi, vi E sI.

(4.23)

K=O,K#I

The second term in Eq. (4.23) represents the aggregation of the complementary subset of macro states into a single external “super” state and the interactions in terms of transition probabilities with this state.

TAKAHASHI’S

ITERATIVE

METHOD

169

Macro-state probability vector Eq. (4.19), together with transition probability aggregation steps as given in Eq. (4.21), constitute one major part of Takahashi’s method, the aggregation step. Micro-state probability vector Eq. (4.23), together with transition probability disaggregations, as indicated in Eq. (4.22), f orm the other major part of Takahashi’s method, the disaggregation step. 4.2.2

Applicability

Given partition: M-l s=

u

SJ,

J=O

a subset Sr is lumpable if and only if real-valued constants condition (4.24) holds Vj E S - SI:

rrj exist such that

ViESI.

pii=rIi

(4.24)

Condition (4.24) is necessary and sufficient for the lumpability of SI with respect to S - SI. The lumpability condition may be restated by referring to Eq. (4.7) if pairwise lumpability is required such that real-valued constants rI J exist and Condition (4.25) holds V’I, J, 0 5 I # J 5 A4 - 1: Vi E SI.

(4.25)

If the state space can be partitioned such that the lumpability condition holds for all pairs of subsets SI, SJ c S, then Takahashi’s algorithm terminates in one iteration with the exact state probability vector V. Unfortunately, lumpability implies very strict structural limitations on the underlying Markov model. More often, however, Markov chains exhibit approximate lumpability such that there exists a sufficiently small lumpability error E: M-l

(4.26)

E= J=O,J#I

iESI

~ESJ

Lumpability error E is a measure for the speed of convergence of Takahashi’s method. The problem remains, however, to find a good partitioning among the states of a given DMTC. Some hints concerning convergence and state space partitioning can be found in Stewart’s book [Stew94]. The underlying Markov models of some so-called product-form queueing networks, which are discussed in Chapter 7, exhibit the exact lumpability property. F’urthermore, the lumpability equations can be exploited theoretically to investigate accuracy of approximate product-form queueing network algorithms.

170

STEADY-STATE AGGREGATION/ DISAGGREGATION METHODS

In general, the efficiency of Takahashi’s method can benefit from an exploitable structure of the underlying transition probability matrix. This fact, however, imposes the burden on the user of finding a good clustering strategy, and no generally applicable rule is known for a good clustering strategy. Finally, it is worth mentioning that approximate lumpability is different from weak lumpability, which can be applied for an aggregation-disaggregation based transient analysis of CTMCs. A comprehensive study on lumpability and weak lumpability is presented by Nicola [NicoSO]. 4.2.3

The Algorithm

Since convergence of Takahashi’s method is a non-trivial problem, some care is needed here. We follow an approach that is often applied to enforce geometric convergence, namely, to incorporate an intermediate power, Gauss-Seidel, or SOR step between successive aggregations and disaggregations. If the decrease in residual error is not sufficient from iteration to iteration, such a step is suggested to be included in the computation. Usually, the corresponding condition is stated in terms of a real-valued lower improvement bound c, 0 < c < 1 on the error ratio from iteration step n - 1 to iteration step n: residualerror (~(“1) residualerror (~(~-l))

(4.27)

’ ‘*

The complete algorithm is presented as Fig. 4.4. 4.2.4

Application

We demonstrate Takahashi’s method with the same example used in Section 4.1 to illustrate Courtois’s method to allow an easy comparison of the numerical behavior of the two algorithms. Recall that we considered a network in which two customers were circulating among three stations according to some stochastic regularities. The resulting state transition diagram of the set S of six states was depicted in Fig. 4.1. Again, we will apply the decomposition strategy indicated in Fig. 4.2 and partition S into three disjoint subsets: So = {020,110,200}, Si = (011, lOl}, and S:! = (002). Arranging the states in the same order as defined in Table 4.3 and using the same numbers in our example results in the following transition probability matrix:

P=

0.52 0.24 0 0.02 0 0 L

0.45 0.265 0.24 0.02 0.02 0

0 0.45 0.745 0 0.02 0

0.03 0.015 0 0.48 0.24 0.02

0 0.03 0.45 0.705

0 (4.28) 0.015

TAKAHASHI’S

ITERATIVE

Create the state space 9 and organize it appropriately according 09 decomposition along the lines of approximate fumpability into probabiJ.ity matrix P = [PJJ~~]. fnitiaLioation: a) r1.:=0; b) estimate Y(O); c> choose c andO
f (w”’

,(nf

function

f([i,

METHOD

to a pattern transition

. .I/);

2 E

= v’P;

n:=n+l;

b)

according to Eq. (4.211; solve yfle) = yfn)I’(nf accordjag

Disa&gregation a)

calculate

for I?$’

all

0 5 15

= &g,

Normalizat With v(n)

to Eq. (4.22); VP’ = furi] with fpP$i

kf -1:

,a, kES1

according b) calculate equations : (n) = & yi

to Eq. (4.19);

tt’j E S Vk

Eq. (4,23)

Fig.

4.4

the system

vi E SIi

+ K=~p?ifYl;)rg. I

foa : = ~(;l’ from the previous E 3

by solving

step solve;

Takahashi’s

v(“)l=l,

algorithm.

of

171

172

STEADY-STATE

AGGREGATiON/DlSAGGREGATlON

METHODS

It has been shown that the transition probability matrix P in Eq. (4.28) is nearly completely decomposable so that Courtois’s method could be applied. We now investigate the lumpability error induced for the same partition with respect to approximate lumpability as defined in Eq. (4.26). To calculate the lumpability error, we first let I = 0, J = 1 and denote with EIJ the inner sum:

601

=

c c xi iESo jESl

- &

c c Pij iESo jESl

= o o3 _ 0.03 + 0.015 + 0.03 + 0.015 3 + 0.015 0.03 - 0.03 + 0.015 + 0.03 + 0.015 + 3 + o 015 _ 0.03 + 0.015 + 0.03 + 0.015 3 = IO.03- 0.031+ 10.045- 0.031+ 10.015- 0.031 = 0 + 0.015 + 0.015 = 0.03. Similarly, we get: E()g= 0. Therefore, by letting I = 0, J # I, the total lumpability error is given by: ql = EOl+ 602= 0.03 + 0 = 0.03. For the other elements of the partition, we get: El = 0.015,

E2= 0.

With: E=

o
= 0.03,

measurescan be derived for the speed of convergence if Takahashi’s method is applied with respect to the given partition of S = U”,=, SI. 4.2.4.1

Aggregation

With the initial probability vector: y(O) = y(0) =

(

111111 $$g’ g’s’ g

the first aggregation step, i.e., the computation of

r(l) = ri:’ [ I

>

7

TAKAHASHI’S

can be accomplished of r$) is shown:

ITERATIVE

METHOD

173

according to Eq. (4.21). As an example, the calculation

00= $ *lxo~ij r(l) = 5 + (0.52 + 0.45 + 0.24 + 0.265 + 0.45 + 0.24 + 0.745) = ; * 2.91 = 0.97. Thus, in the first aggregation sition probability matrix:

I’(‘)

=

step, we get the following

0.97 0.04 0

0.03 0.9375 0.04

0 0.0225 0.96

macro-state

.

tran-

(4.29)

Then the macro-state probability vector y(l) is given by the solution of the following system of linear equations:

#

7p,

#

-0.03 0.04 0

From Eq. (4.30) we obtain decimal digit:

0.03 -0.0625 0.04

0 0.0225 -0.04

the macro-state

= 0,

y(l)1

probabilities

y(l) = (0.4604,0.3453,0.1943)

= 1.

(4.30)

up to the fourth

.

(4.31)

4.2.4.2 Disaggregation In the first part of the disaggregation step, transition probabilities I’if’, are computed:

IT\:) = Jjo.02 + io

= 0.01, r&) = 0.

From there we compute:

(1) (1)+-iil)ral,) (1)-- c vJ(%jo +71 r10 Vo j=o,1,2

= (z@ 0.52 + vi’). 0.24 + v;l’ * 0) + (0.3453.0.01 + 0.1943.0) = (@ - 0.52 + vi’)- 0.24 + z@ 0) + 0.003453.

174

STEADY-STATE

AGGREGATION/

The remaining transition summarized as follows:

DlSAGGREGATlON

probabilities

METHODS

from macro states to micro states are

To complete disaggregation in the first iteration step, the following three sets of equations, for { vir), vi’), Vil)}, {Vi”, Vi” }, and { $)) need to be solved separately for a calculation of micro-state probabilities: VO('I

= ([email protected]

+ $)-0.24)

+ 0.003453,

*1 ('I

= (~$0.45

+ i+O.265

+ [email protected])

V2(l)

= (zp.0

V3 (')

= (z&O.48

+ ~$0.24)

v4 (l)

= (zp.O.45

+ zp.O.705)

V5

(1) _ -

With the solutions dition:

(u5

+ 0.006906,

+ up. 0.45 + up. 0.745) + 0.003453,

(1) ~0.96)

+

+ 0.010792, + 0.010792,

0.00776925.

of these systems

of equations and the normalization

con-

vQ)l = 1, the probability vector Y(‘) at the end of the first iteration to the fourth digit is:

with a precision up

Y(‘) = (0.0812,0.1486,0.2753,0.1221,0.1864,0.1864)

.

(4.32)

In Table 4.5, the micro-state probabilities obtained after one iteration of Takahashi’s method are compared to the results gained if Courtois’s method is applied. The percentage error in the results obtained with Courtois’s approach are also included in the table. It can be seen that even after the first iteration step, the results are already relatively close to the exact values. After four iterations, the results gained with Takahashi’s method resemble the exact values up to the fourth decimal digit. 4.2.5

Final

Remarks

Takahashi’s method was presented in terms of DMTCs.

However this choice

implies no limitation since it is a well-known fact that ergodic DTMCs and

TAKAHASHI’S

Table 4.5

State probabilities:

State

y(O)

y(l)

200 110 020 101 011 002

0.166 0.16g 0.166 0.166 0.166 0.16G

0.0785 0.1436 0.2660 0.1179 0.2138 0.1801

Takahashi’s,

y(2)

y(3)

0.0787 0.1476 0.2770 0.1146 0.2150 0.1672

State

Courtois

Exact

200 110 020 101 011 002

0.0828 0.1480 0.2710 0.1194 0.2108 0.1680

0.0787 0.1478 0.2777 0.1144 0.2150 0.1664

y(4)

0.0787 0.1478 0.2777 0.1144 0.2150 0.1665

0.0787 0.1478 0.2777 0.1144 0.2150 0.1664

ITERATIVE

Courtois’s

175

METHOD

and exact methods

Exact

% Error(‘)

0.0787 0.1478 0.2777 0.1144 0.2150 0.1664

+ +

0.3 2.8 4.2 3.1 0.6 8.2

% Errod4) 0.0 0.0 0.0 0.0 0.0 0.0

% Error + + + +

5.2 0.1 2.4 4.4 2.0 0.1

CTMCs are equivalent with respect to their steady-state probability computations, when applying the transformations given in Eq. (3.6) and Eq. (3.5). But Takahashi’s method can be more conveniently applied directly on the generator matrix Q = [qij] of a CTMC. As usual, we refer to the steady-state probability vector of the given micro-state CTMC through z or VP, respectively. In the continuous-time case,the resulting equations correspond directly to their discrete-time counterparts. Instead of using Eq. (4.19), the macrostate probability vector o is now calculated based on infinitesimal generator matrix Z of the continuous-time macro state process:

o=aq

al=

(4.33)

1.

The matrix entries CIJ of X are calculated with an interpretation corresponding to the one applied to Eq. (4.21):

I # J. The disaggregation steps are analogous Igvi-1: CIj

=

c

iESI

jES,j#i

jESr,j#i

eqij c

kESI

-

to Eq. (4.22)

)

and Eq. (4.23)

j E SJ, J # I,

“g

J=O, J#I

(4.34) VI, 0 5

(4.35)

176

STEADY-STATE

AGGREGATlON/DlSAGGREGATlON

METHODS

Problem 4.5 Compare Courtois’s method to Takahashi’s method for numerical accuracy on the example studies performed in Sections 4.1 and 4.2.4. In particular, explain the observed numerical differences in the corresponding macro-state transition probability matrices r, in the corresponding macrostate probability vectors y, and in the corresponding micro-state probability vectors u. Apply decomposition with regard to station two, as indiProblem 4.6 cated in Fig. 4.3, to the DTMC underlying the discussions in Sections 4.1 and 4.2.4. Build up and organize a transition probability matrix that properly reflects the chosen strategy of decomposition. Calculate the approximate lumpability error E according to Eq. (4.26)) compare it to the results obtained in Section 4.2.4, and interpret the differences. Apply Takahashi’s algorithm on the basis of the decomposiProblem 4.7 tion strategy with regard to station two, as discussed in the previous problem. Compare the resulting macro-state transition probability matrices, the macrostate probability vectors, and the micro-state probability vectors obtained in the iteration step to those obtained under the original decomposition strategy in Section 4.2.4. Comment on and interpret the results. Problem 4.8 Use the modified algorithm of Takahashi and apply it for a numerical analysis of the CTMC specified by its generator matrix Q in Table 4.2 and the parameters in Table 4.1. Apply the same partition as indicated in Table 4.2.

5 Transient Solution of Markov Chains Transient solution is more meaningful than steady-state solution when the system under investigation needs to be evaluated with respect to its shortterm behavior, Using steady-state measures instead of transient measures could lead to substantial errors in this case. Furthermore, applying transient analysis is the onl y choice if non-ergodic models are investigated, Transient analysis of Markov chains has been attracting increasing attention and is of particular importance in dependability modeling. Unlike steady-state analysis, CTMCs and DTMCs have to be treated differently while performing transient analysis. Surprisingly, not many algorithms exist for the transient analysis of DTMCs. Therefore, we primarily focus on methods for computing the transient state probability vector r(t) for CTMCs as defined in Eq. (2.53). Furthermore, additional attention is given to the computation of quantities related to transient probabilities such as cumulative measures. Recall from Eq. (2.53) that for the computation of transient state probability vector n(t), the following linear differential equation has to be solved, given infinitesimal generator matrix Q and initial probability vector m(O):

d7r(t>= m(t)Q, dt

~(0) = (Mom,.

. .> .

(54

Measures that can be immediately derived from transient state probabilities are often referred to as instantaneous measures. However, sometimes measures based on cumulative accomplishments during a given period of time 177

178

TRANSIENT SOLUTION OF MARKOV CHAINS

Fig. 5.1

A pure birth

process.

[0, t) could be more relevant. Let: t

L(t) =

.I

7r(U)dU

0

denote the vector of the total expected times spent in the states of the CTMC during the indicated period of time. By integrating Eq. (2.53) on both sides, we obtain a new differential equation for L(t): -dW

dt

Cumulative

measures

= L(t)Q + r(O),

L(0) = 0.

can be directly computed from the transient solution

of Eq. (5.2).

5.1

TRANSIENT

ANALYSIS

USING EXACT METHODS

We introduce transient analysis with a simple example of a pure birth process. 5.1.1

A Pure Birth Process

As for birth-death processes in steady-state case, we derive a transient closedform solution in this special case. Consider the infinite state CTMC depicted in Fig. 5.1 representing a pure birth process with constant birth rate X. The number of births, N(t), at time t is defined to be the state of the system. The only transitions possible are from state Ic to state k + 1 with rate X. Note that this is a non-irreducible Markov chain for any finite value of X, so the steady-state solution does not exist. From Fig. 5.1, the infinitesimal generator matrix Q is given by:

Due to the special structure of the generator matrix, it is possible to obtain a closed-form transient solution of this process. With generator matrix Q and

TRANSIENT ANALYSIS USING EXACT METHODS

179

Eq. (2.53), we derive a system of linear differential equations for our example:

(5.3) (54 With the initial state probabilities:

‘Irk(O) =

1 Ic=o, 0 k>l, {

(5.5)

the solution of the differential equations can be obtained. From differentiation and integration theory the unique solution of Eq. (5.3) is: 7ro(t) = emAt.

(5.6)

By letting k = 1 in Eq. (5.4) and subsequently substituting for TO(~) into the differential equation, we get:

the solution

$m(t) = -k(t) + xTo(t) = -hrl(t)

+ Xesxt.

Applying again elementary differentiation and integration rules to this different i al equation results in another simple unique solution: nl(t) = AtemAt.

(5.7)

If this process is repeated, we get a closed-form solution for each transient state probability i~~k(t): --Xt,

k > 0 *

The proof of Eq. (5.8) is straightforward using the principle of induction and we leave it as an exercise to the reader. We recognize Eq. (5.8) as the Poisson pmf. Poisson probabilities are plotted as functions of t in Fig. 5.2 for several values of X. Thus, the random variable N(t) at time t is Poisson distributed with parameter X-L,while the stochastic process {N(t)It 2 0) is the Poisson process with rate X. So the transient analysis of a very simple birth process provided as a by-product one of the most important results of Markov chain and queueing theory, i.e., the stochastic process under consideration is the Poisson process. One important measure of the Poisson process is its mean value function m(t), defined as the expected number of births in the

180

TRANSIENT SOLUTION OF MARKOV CHAINS

1

0.8

fig. 5.2

Selected Poisson probabilities

with parameters X = 0.5 and 1.0.

TRANSIENT ANALYSIS USING EXACT METHODS

181

interval [0, t). This measure is computed as: m(t) = 2 i7r$) = 2 i y+ i=o * i=o

= e-xt 2 -(At) izl (i(W1 - v

= (Xt)cAt 2 y i=o *

(5.9)

= (Xt)emxtex” = At. Eq. (5.8) together with Eq. (5.9) characterize the Poisson pmf and its mean At. Problem 5.1 Using the principle of mathematical induction, show that the Poisson pmf as given in Eq. (5.8) is the solution to Eq. (5.4). Problem 5.2 Derive the solution of Eq. (5.2) of the pure birth process of Fig. 5.1, assuming no(O) = 1 and ~(0) = 0, V’k > 1. 5.1.2

A Two-State

CTMC

While in the previous section transient analysis based on closed-form expressions was possible due to very simple and regular model structure, in this section we carry out closed-form transient analysis of a CTMC with two states. Our treatment closely follows that in [STP96]. Fig. 5.3 shows a homogeneous, continuous time Markov chain model for this system with state space S = (0, 1). A typical application of this model is to a system subject to failure and repair.

Fig. 5.3

A simple two-state CTMC.

The generator matrix of the CTMC of Fig. 5.3 is:

Q=[

;’

:A].

From the system of Eqs. (2.51), we get:

&o(t) = -P-o(t) + h(t), d p(t)

= wo(t>

-

h(t).

(5.10)

182

TRANSIENT SOLUTION OF MARKOV CHAINS

Applying the law of total probability, To@>

we conclude:

=

1 -

w(t),

and rewrite Eq. (5.10) as: $1(t)

= p(1 - 7r1(t))

- h(t)

or: d g1

(t> + (P + +h(q

= p.

A standard method of analysis for linear differential Eq. (5.11) is to use the method of integrating factors [RaBe74]. Both sides of the (rearranged) equation are multiplied by the integrating factor:

to get:

p&+W = &+W --gl(t) d + (p + X)e(p+x)trl(t)

(5.12) (5.13)

By inspection of Eq. (5.12) it can be seen that the sum expression on the right side of the equation equals the derivative of the product of the subterms so that Eq. (5.13) results. Integrating Eq. (5.13) on both sides gives us as an intermediate step: I-L -e(CL+‘Jt PL+X Multiplying

+ c = e(P+‘b

71‘1(t).

(5.14)

Eq. (5.14) with: ,++w

provides the desired result: w(t)

=

5

+ ce-(P+X)t

(5.15)

Integration constant c reflects the dependency of the transient state probabilities on the initial probability vector m(O). Assuming, for example: 7r(l)(O) = (0,l) , results in: 1 = r(l)1 (0) =

5

=pL+c*

+ ce-(P+x)o

TRANSIENT ANALYSIS USING EXACT METHODS

Thus, unconditioning integration constant:

with an initial

c=l-

183

pmf 7r(l)(O) yields the corresponding

-I- P Pu+X

x p+X’

The final expressions of the transient state probabilities

in this case results

in:

x (,++X)t) ) @(t)=Lp+x+p+x 7&t> =1- 7p(t)

(5.16)

p + X (e--(p+X)t) =-- p + X I-L+X ,w+X = X - X (e--(p+X)t) PL+X

= - x - X (e-(P+h)t) . p+x p+x With a closed-form expression of the transient state probabilities formance measure can be easily derived.

(5.17) given, per-

Problem 5.3 First, derive expressions for the steady-state probabilities of the CTMC shown in Fig. 5.3 by referring to Eq. (2.58). Then, take the limits as t -+ 00 of the transient pmf z(‘)(t) and compare the results to the steady-state case 7r. Problem 5.4 Derive the transient state probabilities of the CMTC shown in Fig. 5.3 assuming initial pmf rrc2)(0) = (0.5,0.5). Take the limit as t + 00 of the resulting transient pmf 7rc2)(t) and compare the results to those gained with 7r(‘)()) when taking the limit. Problem 5.5 Assume the system modeled by the CTMC in Fig. 5.3 to be “up” in state 1 and to be “down” in state 0. (a) Define the reward assignments for a computation of availability measures based on Table 2.1. (b) Derive formulas for the instantaneous availability A(t), the interval unavailability UA( t) , and the steady-state availability A for this model, based both on 7r(‘)(t) and 7rc2)(t). Refer to Section 2.2.2.3.1 for details on how these availability measures are defined. (c) Let p = 1, X = 0.2 and compute all measures that have been specified in this problem. Evaluate the transient measures at time instants t E {1,2,3,5,10>.

186

TRANSIENT SOLUTION OF MARKOV CHAINS

computations with small numbers, a left truncation point 1 can be introduced. To this end an overall error tolerance E = ~1+E, is partitioned to cover both left and right truncation errors. Applying again a vector norm, a left truncation point 1 can be determined similarly to the right truncation point r:

1-l c

e-qt -(qtY

i=O

i!

5 El.

(5.24)

With appropriately defined truncation points I and r as in Eqs. (5.23) and (5.24), the original uniformization Eq. (5.21) can be approximated for an implement at ion: 7r(t) 73 2 v(i)emqtq, i=l

.

Y(0) = 7T(O).

(5.25)

In particular, O(a) terms are needed between the left and the right truncation points and the transient DTMC state probability vector Y(Z) must be computed at the left truncation point 1. The latter operation according to Eq. (5.22) requires O(qt) computation ‘time. Thus the overall complexity of uniformization is 0 (qqt) [ReTr88]. (Here q denotes the number of nonzero entries in Q.) To avoid underflow, the method of Fox and Glynn can be applied in the computation of Z and r [FoG188]. Matrix squaring can be exploited for a reduction of computational complexity [ReTr88, AbMa93]. But because matrix fill-ins might result, this approach is often limited to models of medium size (around 500 states). 5.1.4.2 Stiffness Tolerant Uniformiza tion As noted, uniformization is plagued by the stiffness index qt [ReTr88]. Muppala and Trivedi observed that the most time consuming part of uniformization is the iteration to compute y(i) in Eq. (5.22). But this iteration is the main step in the power method for computing the steady-state probabilities of a DTMC as discussed in Chapter 2. Now if and when the values of y(i) converge to a stationary value ti, the iteration Eq. (5.22) can be terminated resulting in considerable savings in computation time [MuTr92]. We have also seen in Chapter 2 that speed of convergence to a stationary probability vector is governed by the second largest eigenvalue of the DTMC, but not by qt. However, since the probability vectors v(i) are computed iteratively according to the power method, convergence still remains to be effectively determined. Because an a priori determination of time of convergence is not feasible, three cases have to be differentiated for a computation of the transient state probability vector z(t) [MuTr92, MMT94] :

1. Convergence occurs beyond the right truncation point. In this case, computation of a stationary probability vector is not effective and the transient state probability vector ?r(t) is calculated according to Eq. (5.25) without modification.

188

TRANSIENT SOLUTION OF MARKOV CHAINS

1. Convergence occurs beyond the truncation point, that is, c > r. As a consequence, Eq. (5.29) remains unaffected and must be evaluated as is. 2. If convergence occurs before truncation point, Eq. (5.29) is adjusted to: L(t) = k fy v(i)L’(t) 2=0 = f 5 v(i)L’(t) 2=0

+ TV

= ; $ v(i)L’(t) a=0

+ fv(c)

= ig

2

L’(t)

kc+1

2 L’(i) - 2 L’(t) i=O ( i=o

u (38 L ‘( t >+$44 a=0

)

(91-@f(t))

= ; 2 u(i) (l-ge-qq i=o iqt-g

+$44

(1-$0tQ$)),

(5.30)

with

In case truncation needs to be performed, a time-dependent error-bound estimate d’)(t) related to Eq. (5.29) can be given as a function of the truncation point r. Assuming an error tolerance allowance, the corresponding number of required terms r can be determined:

&9(t) 5 I 2

2

e-Gy

q i=r+1j=i+1 5;

,g

’

(i-(r+l))e-qty

z=?-+1 i

< -

l

cc

ie-ql(;) --f

c G i=r+1

&cqtll--

i==T

,e -

z=r+1 r+l

(d

.

(r+l)e-d$

O3 q

c i=r+l

(?-qt Glv i!

*

(5.31)

TRANSIENT ANALYSIS USING EXACT METHODS

5.1.5

Other

Numerical

189

Methods

We briefly discuss numerical methods based on ordinary differential equation approach followed by methods exploiting weak lumpability. 5.1.5.1 Ordinary Differential Equations Standard techniques for the solution of ordinary differential equations (ODE) can be utilized for the numerical solution of the Kolmogorov differential equations of a CTMC. Such ODE solution methods discretize the solution interval into a finite number of time intervals {ti, 62,. . *, t’Z?“‘? t 7% }. The difference between successive time points, called the step size h, can vary from step to step. There are two basic types of ODE solution methods: explicit and implicit. In an explicit method, the solution rr(ti) is approximated based on values 7r(tj) for j < i. The computational complexity of explicit methods such as Runge-Kutta is O(qqt) [ReTr88]. Although explicit ODE-based methods generally provide good results for non-stiff models, they are inadequate if stiff models need to be studied. Note that stiff CTMCs are commonly encountered in dependability modeling. In an implicit method, r(ti) is approximated based on values n;TT( tj) for j 5 i. Examples of implicit methods are TR-BDF2 [ReTr88] and implicit Runge-Kutta [MMT94]. At each time step a linear system solution is required in an implicit method. The increased overhead is compensated for by better stability properties and lower computational complexity on stiff models [MMT94, ReTr88]. For non-stiff models, Uniformization is the method of choice, while for moderately stiff models, uniformization with steady-state detection is recommended [MMT94]. For extremely stiff models, TR-BDF2 works well if the accuracy required is low (eight decimal digits). For high accuracy on extremely stiff models, implicit Runge-Kutta is recommended [MMT94].

5.1.5.2 Weak Lumpability An alternative method for transient analysis based on weak lumpability has been introduced by Nicola [NicoSO]. Nicola’s method is the transient counterpart of Takahashi’s steady-state method, which was introduced in Section 4.2. Recall that state lumping is an exact approach for an analysis of a CMTC with reduced state space and, therefore, with reduced computational requirements. State lumping can be a very efficient method for a computation of transient state probabilities, if the lumpability conditions apply. Note that state lumping can also be orthogonally combined with other computational methods of choice, such as stiff uniformization, to yield an overall highly accurate and efficient method. For the sake of completeness, we present the basic definitions of weak lumpability without going into further detail here. Given conditional proba-

192

TRANSIENT SOLUTION OF MARKOV CHAINS

A00 Aol ...

previously described state partition:

41

Al0

Q=

.

;

L

A(F-1)0 AFO

. . .

e-S ...

QF-1 AF(F-1)

AOF &F ..

(5.35)

.

A(F-l)F AFF*i

Let Qr = Q~r,l 5 I 5 F - 1, denote submatrices containing transition rates between grouped states within fast recurrent subset Sr only. Submatrices AIJ, I # J, contain entries of transitions rates from subset SI to SJ. Finally, Ace and AFF collect intra slow states and intra fast transient states transition rates, respectively. Matrices Qr , 1 5 I 5 F - 1, contain at least one fast entry in each row. Furthermore, there must be at least one fast entry in one of the matrices AFI, 0 5 I 5 F - 1 if SF # 8. Matrix AFF may contain zero or more fast entries. Finally, all other matrices, i.e., Aoo,&J,-~ # J,O L I L F-LO 5 J < F, cant ain only slow entries. By definition, only slow transitions are possible among these subsets SI and SJ. An approximation to r(t) is derived, where n(t) is the solution to the differential Eq. (5.36):

The reorganized matrix Q forms the basis to create the macro-state generator matrix X. Three steps have to be carried out for this purpose: first, aggregation of the fast recurrent subsets into macro states and the corresponding adaptation of the transition rates among macro states and remaining fast transient states, resulting in intermediate generator matrix %. In a second step, the fast transient states are eliminated and the transition rates between the remaining slow states are adjusted, yielding the final generator matrix X. Finally, the initial state probability vector ~(0) is condensed into a(O) as per the aggregation pattern. Transient solution of differential Eq. (5.37) describing the long-term interactions between macro states is carried out:

$7(t) = tT(t)lx, g(o)

=

(5.37)

(~Oo(~),~~~,~O,,_,(~),~l(~),~~~,~F-l(~))

a

Once the macro state probability vector o(t) has been computed, disaggregations can be performed to yield an approximation r”(t) of the complete probability vector n(t). 5.2.2

Aggregation

of Fast Recurrent

Subsets

Each subset of fast recurrent states is analyzed in isolation from the rest of the system by cutting off all slow transitions leading out of the aggregate. Collecting all states from such a subset S’I and arranging them together with the

AGGREGATION OF STIFF MARKOV CHAINS

193

corresponding entries of the originally given infinitesimal generator matrix Q into a submatrix &I gives rise to the possibility of computing the conditional steady-state probability vectors XT, 1 < I 5 F - 1. Since &I does not, in general, satisfy the properties of an infinitesimal generator matrix, it needs to be modified to matrix QT:

Q;=QI+DI.

(5.38)

Matrix DI is a diagonal matrix whose entries are the sum of all cut-off transition rates for the corresponding state. Note that these rates are, by definition, orders of magnitude smaller than the entries on the diagonal of Qr. The inverse of this quantity is used as a measure of coupling between the subset Sr and the rest of the system. Since Q: is the infinitesimal generator matrix of an ergodic CMTC, the solution of: $Q;

7rp = 1

= 0,

(5.39)

yields the desired conditional steady-state probability vector X; for the subset of fast states SI. For each such subset SI, a macro state I is defined. The set of aggregated states {I, 1 5 I 5 F - I}, constructed in this way, together with the initially given slow states Se form the set of macro states niir = SoU{I, 1 5 I 5 F - 1) for which a transient analysis is performed in order to account for long-term effects among the set of all macro states. To create the intermediate generator matrix 2 of size (]1M( + no) x (]A!!] + no), unconditionings and aggregations of the transition rates have to be performed. The dimensions [no x 7251of submatrices 21~ are added to the corresponding equations in what follows. Let us define matrices of appropriate size: 1 ... 1 ...

1

E=

containing 1s only as entries. The following cases are distinguished: l

Transitions remain unchanged among slow states in Se:

zoo= Aoo, [nox no]*

(5.40)

0 nanSitiOnS involving fast transient states from SF: - Transitions between slow states and fast transient states, and vice versa, remain unchanged: EFO = AFO,

[nF x 72017

(5.41)

EOF =AoF,

[no x nF].

(5.42)

195

AGGREGATION OF STIFF MARKOV CHAINS

a way that it reflects the partitioning states:

into macro states and fast transient

(5.49) If no fast transient states are included, then X = Z already constitutes the final generator matrix. In any case, a transient analysis can be carried out based on a model with reduced state space. Although the intermediate model may still be stiff because fast and slow states are still simultaneously included, the computational savings can be considerable due to smaller state space. But more aggregation is possible, as is highlighted in the following section. With 2 given, a differential equation can be derived for the computation of the transient state probability vector 3(t), conditioned on initial probability vector 8(O):

$=(t)= 5(t)%, w> 5.2.3

(5.50)

= (km..

. ,~o,,~Jo>,~l(o),

. . * ,&?-1(0),aF,(0),

Aggregation

of Fast Transient

Subsets

. . . ,i?F,F-l(o))

.

Recall that a fast transient subset of states is connected to states in other subsets by at least one fast transition. Thus if such a set is not nearly completely decomposable from the rest of the system, it is tightly coupled to at least one other non-empty subset of states. Now consider an intermediate state space consisting of a non-empty fast transient subset of states SF c S, and the set of macro states n/r = SoU{I, 1 5 I 5 F - l} resulting from aggregation of fast recurrent subsets and collecting them together with the slow states. Furthermore, assume the intermediate transition rates among the states in MU SF to be already calculated as shown in the previous section. From Eq. (5.50) and Eq. (5.49)) the following two equations can be derived: (5.51) (5.52) It is reasonable to assume that for the fast transient states, the derivative on the left-hand side of Eq. (5.52) approaches zero (i.e., d/d-G-~(t) = 0), with respect to the time scale of slow states. Hence from Eq. (5.52) we obtain an approximation of 0~ :

196

TRANSIENT SOLUTION OF MARKOV CHAINS

Using the result from Eq. (5.53) in Eq. (5.51) provides us with the following approximation:

(5.54) = c+gf(t) (2

MM

-

~M&%M

= +z(t)k;,.

>

(5.55) (5.56)

Hence, E x %EM is taken as the infinitesimal generator matrix of the final aggregated Markov chain. Bobbio and Trivedi [BoTr86] have shown that:

(5.57) is the asymptotic exit probability matrix from the subset of fast transient states SF to macro states M. The matrix entries of PFM = bi~(O, oo)] ,i E SF, I E AI are the conditional transition probabilities that the stochastic process, once initiated in state i at time t = 0 will ultimately exit SF and hit macro state I E 44 as time t -+ co. These probabilities are used to adjust the transition rates among macro states during elimination of fast transient states. The interpretation is that the fast transient states form a probabilistic switch in the limit, and the time spent in these sets of states can be neglected in the long run. Furthermore, related to this assumption, the exit stationary probability will be reached in a period of time much smaller than the time scale characterizing the dynamics among the macro states. The error induced by this approximation is inversely proportional to the stiffness ratio. For the sake of completeness, it is worth mentioning that the entries in matrix I!ZMF from Eq. (5.55) denote exit rates, leading from macro states in AL!via the probabilistic switch, represented by matrix PFM Eq. (5.57),_back to macro states. Hence, the rate matrix YZMFPFM is simply added to XMM in Eq. (5.55) to obtain the rate adjustment. 5.2.4

Aggregation

of Initial

State

Probabilities

Since the transient analysis is performed only on the set of macro states M, the initial probability vector ~(0) must be adjusted accordingly. This adjustment is achieved in two steps. First, the probability mass z1(0) assigned to micro states i E ,571,which are condensed into a single macro state I, is accumulated for all I, 1 5 I 5 F - 1:

61(O) = 7rI(O)l.

(5.58)

Second, if SF # 8, then the probability mass ?TF(O) assigned to the fast transient states SF is spread probabilistically across the macro states A4 according to rule implied by the probabilistic switch PFM given in Eq. (5.57):

5;(O) = &(O)+TF(O)PFM.

(5.59)

AGGREGATION OF STIFF MARKOV CHAINS

197

Depending on SF, the initial probability vector a(O) is given by 5; (0) according to Eq. (5.58) or by 5(O) according to Eq. (5.59). The aggregation part has now completely been described. With this technique, the transient analysis of large stiff Markov chains can be reduced to the analysis of smaller non-stiff chains. Transient analysis is accomplished only for the macro states in the model, be it either the initially present slow states or the macro states resulting from an aggregation of fast recurrent subsets of states. The accuracy of the method is inversely proportional to the stiffness ratio of the initial model. With the assumption that steady state is reached much quicker for the fast states than for the slow states, further approximations become possible. Applying certain disaggregation steps, approximations of the transient fast state probabilities can also be derived, thus resulting in a complete state probability vector. 5.2.5

Disaggregations

525.1 Fast Transient States With the approximate macro state probability vector &E(t), obtained as the solution of Eq. (5.56), the approximate fast transient state probability vector 8:(t) can be derived easily with a transient interpretation of Eq. (5.53):

*F(t) = -c&(t)&&;;.

(5.60)

With:

c = b;(t)l+

iqtp,

(5.61)

normalization can be taken into account for computation of a probability vector so that the first disaggregation step is accomplished, leading to the intermediate transient state probability vector u- =(c) (t) as an approximate solution of Eq. (5.50): b(t) z ec)

(t) = i (i?%(t), a;(t))

.

(5.62)

5.2.5.2 Fast Recurrent States In a second disaggregation step, the transient micro-state probability vector m”(t) is also calculated as an approximation of n(t), which is formally defined as the solution of Eq. (5.36), incorporating the original infinitesimal generator matrix Q from (5.35). For a computation of 7?(t) = @y(t)), we need to refer to the conditional micro-state probability vectors 7r; = (7rT0,.. . ,7r;,1P1 ), 1 5 I -< F - 1, from Eq. (5.39). The transient macro-state probabilities ar(t), 1 5 I 5 F - 1, are used to uncondition the conditional steady-state probability vectors $, 1 5 I 5 F - 1: 7rr(t) z 7ry (t) = (ar(t)-$)

.

(5.63)

198

TRANSIENT SOLUTION OF MARKOV CHAINS

If SF # 0 unconditioning has to be performed on the basis of b(t) &“)(t) according to Eq. (5.62):

7Q(t) x Try (4 = ( p

(4

$

) .

z

(5.64)

Collecting all the intermediate results from Eq. (5.62) and Eq. (5.64) yields the final approximate transient probability vector in its most general form:

=

(

p(t),Ty(t),

. . . ,7r~-l(t),ti~~c~(t))

.

(5.65)

If SF = 8 then Eq. (5.65) simplifies to the following:

7(t) E57P (t> = (ab&), . * * 7~o,,-l (4,

(5.66) 5.2.6

The Algorithm

Initialization: l

Specify r >> l/T, the transition rate threshold, as a function of the time horizon T for which the transient analysis is to be carried out.

l

Partition the state space S into slow states SO and fast states S - SO with respect to r.

l

Partition the fast states S - So further into fast recurrent subsets Sl, 1 5 I < F - 1 and possibly one subset of fast transient states SF.

l

Arrange the infinitesimal generator matrix Q according to Eq. (5.35). Steady-state analysis of fast recurrent subsets:

l

l

Prom submatrices Q1 (Eq. (5.35)), 0bt ain infinitesimal ces Q; according to Eq. (5.38). Compute the conditional micro state probability

generator matri-

vectors $ according to

Construction of the intermediate generator matrix 2 as in Eq. (5.49): l

Use Q from Eq. (5.35) and n;, 1 5 I 5 F - 1 from Eq. (5.39) and apply operations starting from Eq. (5.40) through Eq. (5.48).

AGGREGATION OF STIFF MARKOV CHAINS

199

Aggregation of fast transient states: IF SF # 8

l

- THEN perform aggregation of fast transient states and construct -E;M: * Compu_te the probabilistic switch PFM according to Eq. (5.57) and ELM according to Eq. (5.55) and Eq. (5.56) from 2. * x=%~&f. - ELSE X = 2:. Aggregation of the initial state probability

vector a(O):

l

Accumulate initial probabilities ~1, (0) 0f micro states i E SI into 61(O), 1 5 Is F - 1 according to Eq. (5.58).

l

IF SF # 8 - THEN calculate approximate initial macro-state probability o(0) = 5$(O) according to Eq. (5.59).

vector

- ELSE a(O) = k(O) according to Eq. (5.58). Computation of the transient macro-state probability l

vector a(t):

Solve Eq. (5.37). Disaggregations:

- THEN * Compute intermediate transient state probability according to Eq. (5.62).

vector 6(t)

* Compute the approximate micro-state probability subvectors n?(t) by unconditioning of $ according to Eq. (5.64) for all 1 < I 5 F - 1. - ELSE compute ?ry(t) by unconditioning of 7rT according to Eq. (5.63) for all 1 5 I 5 F - 1. Final result: l

Compose the approximate transient probability vector r(t) according to Eq. (5.65) if SF # 0 or Eq. (5.66) if SF = 8.

w n”(t)

200

TRANSIENT SOLUTION OF MARKOV CHAINS

5.2.7

An Example:

Server

Breakdown

and Repair

Assume a queueing system with a single server station, where a maximum of m customers can wait for service. Customers arrive at the station with arrival rate X. Service time is exponentially distributed with mean I/,X Furthermore, the single server is subject to failure and repair, both times to failure and repair times being exponentially distributed with parameters y and 6, respectively. It is reasonable to assume that X and /J differ by orders of magnitude relative to y and 6. Usually, repair of a failed unit takes much longer than traffic-related events in computer systems. This condition is even more relevant for failure events that are relatively infrequent. While traffic-related events typically take place in the order of micro seconds, repair durations are in the order of minutes, hours, or days, and failure events in the order of months, years, or multiple thereof. Thus, the transition rates X and p can be classified as being fast, and y and S as slow. Note that we are interested in computing performance measures related to traffic events so that for the rate threshold, the relation r < S < y generally holds and the application of the aggregation/disaggregation approach is admissible. The described scenario is captured by a CTMC with a state space S = {(M),O L 1 I m,k E {OJ}}, w h ere Z denotes the number of customers in the queue and Ic the number of non-failed servers, as depicted in Fig. 5.4a. The state space S is partitioned according to the classification scheme applied to the rates, into a set of slow states Se, a set of fast recurrent states Si, and a set of fast transient states SF = S2: = {(m,W, Sl = {(O, I>, (1,1>, -. . , Cm, I)),

SO

S2 = {(0,0),(1,0),-,(m-

LO)}.

For the case of m = 2, the following state ordering is used in compliance with the partitioning:

&

lo

11

(071)

(171)

20

12 c&l>

W)

21 (LO)

*

To complete initialization, the infinitesimal generator matrix Q, m = 2, is restructured accordingly to reflect the partitioning:

Q=

-6 0 0 Y 0 x

-(x0+?) P 0 6 0

0 x -(X+Y+P) P 0 s

s 0 -(cw 0 0

0 Y

0 0

t

Y 0

-(A + S) 0

-(AA+

6)

. 1

202

TRANSIENT SOLUTION OF MARKOV CHAINS

birth-death process as described by QT : lq =

0-x i CL

l-; l-

3’

0

i = 0, 1, . . . ) 2.

;

To aggregate the set of fast recurrent states Sr into macro state 1, $ is used to derive the intermediate infinitesimal generator matrix 2 for the reduced state space {(2,0), 1, (O,O), (1,O)). By applying Eq. (5.40) through Eq. (5.48):

g=

-4 G,r 0 x

s -Y s 6

0 G,Y -(X+6) 0

0 xi,? x -(A + S)

Matrix 2 represents the CTMC as depicted in Fig. 5.4b for the case of and the m = 2. For an aggregation of the fast transient states, first 2,; probabilistic switch P FM are calculated from 9 according to Eq. (5.57):

With 2 and PFM given, I= = 2”MM can be easily derived using some algebraic manipulation according to Eq. (5.55) and Eq. (5.56):

where X/(X + S) d enotes the probability that a new customer arrives while repair is being performed, and y (X/(X + s>)2-i is the probability that a failure occurs when there are i customers present and that m - i customers arrive during repair. An initial probability vector ~(0) = (0, 1, 0, . . . , 0) is assumed, i.e., ~1~ = 1. This condition implies a(0) = (0,l) in Eq. (5.58) and Eq. (5.59). From Eqs. (5.16) and (5.17) we know symbolic expressions for the transient state probabilities of a two-state CTMC, given an initial pmf r(O). We can therefore apply this result by substituting the rates from I% into these expressions. By letting: 2

cl=ypri i=o

(

x xs6

2-i

)

’

AGGREGATION OF STIFF MARKOV CHAINS

203

we get the following macro-St ate probabilities: ao(t> = 1 - a(t), s m(t)

=

G

+

&

(,++a)

.

Finally, disaggregations must be performed. Given &c(t) = a(t), Eq. (5.60) needs to be evaluated for a computation of the fast transient probability vector:

i m(t)~dm&j

=

(QA+6

+$J).

(

Next, normalization

is accomplished with help of c according to Eq. (5.61):

c= 1+01(t)&

(6

(1+&J

The intermediate transient state probability lows immediately : W) = ;

(

1 - ol(t),~l(t),~l(t)~,~l(t)~

vector a(t) from Eq. (5.62) fol-

i

The final step consists of unconditioning $: = -01 w (TTo’ C

7qt)

+c).

(Qx + 6 +r:l)). the micro-state probability

4,)

vector

7r*12 ) .

Collecting all the results provides an approximation m%(t) to the transient probability vector, which is represented as the transpose:

7r(t)

73 n”(t)

=

204

TRANSIENT SOLUTION OF MARKOV CHAlNS

From the transient state probabilities, all the performance measures of interest can be calculated. Table5.2 Parameter set for Table 5.2. 6

Y

x

I-1

0.01

0.0001

0.5

1

The exact transient state probability vector n(t) is compared to the approximate state probability vector m”(t) at time instants t = 10,100 and 1000 in Table 5.2, with the parameter set from Table 5.1. (Note that the initial probability vectors are chosen as has been specified earlier.) The results of the approximate method are generally very close to the exact values for most states. Usually, the accuracy depends on the length of the investigated time horizon for a transient analysis. Short-term results are particularly dependent on the initial probability vector so that some more significant differences can arise as in the case of state (2,O) in our example, where the absolute state probabilities are rather small, and hence the percentage difference is high. Furthermore, whether the error is positive or negative may change over time.

0.008

0.006

0.004

0.002 8 A Q 5 2 2

0

-0.002

-0.004

-0.006

-0.008

Fig. 5.5

I I

I

0.0001

0.001

Errors of the approximate method as a function y.

#I

0.01

Y

AGGREGATION

OF STIFF

MARKOV

205

CHAINS

State probabilities as a function of time: exact vs. approximate

Table 5.2

Exact r(t) No.

State

0 lo

ii>;;

11 12

(1:1) Gw

20

(ON

21

(170)

t=o 0 1 0 0 0 0

t = 10

t = 100

t = 1000

0.00067567 0.57099292 0.28540995 0.14264597 0.00011195 0.00016354

0.00601869 0.56777268 0.28392083 0.14201127 0.00011134 0.00016484

0.00962544 0.56567586 0.28289339 0.14152880 0.00011092 0.00016421

Approx zTM(t) No.

State

0 lo

cw>

t=o 0 0.571429 0.285714 0.142857 0 0

11 12

20 21

t = 10

t = 100

t = 1000

0.00092448 0.57074200 0.28537100 0.14268500 0.00011191 0.00016567

0.00611894 0.56777400 0.28388700 0.14194400 0.00011133 0.00016481

0.00962543 0.56577100 0.28288600 0.14144300 0.00011094 0.00016423

Absolute error No. 0 lo 11 12

20 21

State

t

= 10

-2.488e-04 2.509e-04 3.895e-05 -3.903e-05 4e-08 -2.13e-06

t = 100 -l.O03e-04 -1.32e-06 3.383e-05 6.727e-05 le-08 3e-08

Error %

t

= 1000

le-08 -9.514e-05 7.39e-06 8.58e-05 -2e-08 -2e-08

t

= 10

-36.8242 0.04394 0.01365 -0.02736 0.03573 -1.30243

t

= 100

-1.66564 -0.00023 0.01192 0.04737 0.00898 0.01820

t

= 1000 0.00010 -0.01682 0.00261 0.06062 -0.01803 -0.01218

The impact of different degrees of stiffness on the accuracy of the results are indicated in Table 5.3 and in Fig. 5.5. In particular, the empirical results depicted in Fig. 5.5 support the assumption of the approximation being better for stiffer models. Almost no error is observed if y 5 10v4, while the error substantially increases for the less stiff models as y gets larger. The results depend on the three different parameter sets given in Table 5.4. A closer look at Fig. 5.5 reveals a non-monotonic ordering of the degree of accuracy. Let y = 0.01, then the worst result is obtained for the largest 6, that is, S = 0.1 in state (2, l), as anticipated. But this behavior does not occur for y < 0.001, where the approximate probability is closer to the exact value for S = 0.1 than for 6 = 0.01. Similar patterns can be observed in other cases, as well. Problem 5.6 Verify the construction of the intermediate infinitesimal generator matrix & in the example study by applying Eq. (5.40) through Eq. (5.48) for each submatrix of Q in the indicated way.

AGGREGATION OF STIFF MARKOV CHAINS

Table 5.4 Experiment ;i; (3)

207

Parameter set for Table 5.3 6

Y

x

I-L

0.001 0.01 0.1

0.0001 0.001 0.01

0.5

1

full, in which case they must be rejected. Use this probability to calculate the eflective arrival rate X,, defined as X minus the rejected portion of arrivals. Use Little’s law to calculate the average response time. Apply the aggregation technique for the transient analProblem 5.10 ysis of stiff Markov chains of this section to the example in Section 4.1 and derive the approximate transient probability vector. Assume ~(0) = 111111 transient state probability vector g7 g ), evaluate the approximate ( g7 g’g’g, at time instances t E {1,2,3,5, lo}, and compare the results to those obtained by applying Courtois’s method and to the exact results.

6 Single Station

Queueing Systems

A single station queueing system, as shown in Fig. 6.1, consists of a queueing buffer of finite or infinite size and one or more identical servers. Such an elementary queueing system is also referred to as a service station or, simply, as a node. A server can only serve one customer at a time and hence, it is

Arriving Jobs

Departing Jobs

Servers

Fig. 6.1

Service station with m servers (a multiple server station).

either in a “busy” or an “idle” state. If all servers are busy upon the arrival of a customer, the newly arriving customer is buffered, assuming that buffer space is available, and waits for its turn. When the customer currently in service departs, one of the waiting customers is selected for service according to a queueing (or scheduling) discipline. An elementary queueing system is further described by an arrival process, which can be characterized by its sequence of interarrival time random variables {Al, As, . . s}. It is common to assume that the sequence of interarrival times is independent and identically distributed, leading to an arrival process that is known as a renewal process [Triv82]. Recent work on correlated interarrival times can be found in [LucaSl] and [Luca93]. The distribution function of interarrival times can be continuous or discrete. We deal only with the former case in this book. 209

210

SINGLE STATION QUEUEING SYSTEMS

For information related to discrete interarrival time distributions, the reader may consult recent work on discrete queueing networks [Dadu96]. The average interarrival time is denoted by E[A] = TA and its reciprocal by the average arrival rate X:

The most common interarrival time distribution is the exponential, in which case the arrival process is Poisson. The sequence {Bi, &, +. +} of service times of successive jobs also needs to be specified. We assume that this sequence is also a set of independent random variables with a common distribution function. The mean service time E[B] is ’ d enoted by TB and its reciprocal by the service rate ,X

(6.2) 6.1

KENDALL’S

NOTATION

The following notation, known as Kendall’s notation, is widely used to describe elementary queueing systems: A/B/m

- queueing discipline,

where A indicates the distribution of the interarrival times, B denotes the distribution of the service times, and m is the number of servers (m 2 1). The following symbols are normally used for A and B: M EI, Hk Ck D G GI

Exponential distribution (memoryless property) Erlang distribution with Ic phases Hyperexponential distribution with k phases Cox distribution with k phases Deterministic distribution, i.e., the interarrival time or service time is constant General distribution General distribution with independent interarrival times

Due to proliferation of high-speed networks, there is considerable interest in traffic arrival processes where successive arrivals are correlated. Such non-G1 arrival processes include Markov modulated Poisson process (MMPP) [FiMe93] or batch Markovian arrival process (BMAP) [LucaSl]. Queueing systems with such complex arrival processes have also been analyzed [Luca93]. Except for an example of the use of MMPP, we do not consider such queueing systems in this book. The queueing discipline or service strategy determines which job is selected from the queue for processing when a server becomes available. Some commonly used queueing disciplines are:

KENDALL’S NOTATION

211

FCFS (First-Come-First-Served): If no queueing discipline is given in the Kendall notation, then the default is assumed to be the FCFS discipline. The jobs are served in the order of their arrival. LCFS (Last-Come-First-Served): SIR0 (Service-In-Random-Order): random.

The job that arrived last is served next. The job to be served next is selected at

RR (Round Robin): If the servicing of a job is not completed at the end of a time slice of specified length, the job is preempted and returns to the queue, which is served according to FCFS. This action is repeated until the job service is completed. PS (Processor Sharing): This strategy corresponds to round robin with infinitesimally small time slices. It is as if all jobs are served simultaneously and the service time is increased correspondingly. IS (Infinite Server): There is an ample number of servers so that no queue ever forms. Static Priorities: The selection depends on priorities that are permanently assigned to the job. Within a class of jobs with the same priority, FCFS is used to select the next job to be processed. Dynamic Priorities: The selection depends on dynamic priorities that alter with the passing of time. Preemption: If priority or LCFS discipline is used, then the job currently being processed is interrupted and preempted if there is a job in the queue with a higher priority. As an example of Kendall’s notation, the expression M/G/l

- LCFS preemptive resume (PR)

describes an elementary queueing system with exponentially distributed interarrival times, arbitrarily distributed service times, and a single server. The queueing discipline is LCFS where a newly arriving job interrupts the job currently being processed and replaces it in the server. The servicing of the job that was interrupted is resumed only after all jobs that arrived after it have completed service. Kendall’s notation can be extended in various ways. An additional parameter is often introduced to represent the number of places in the queue (if the queue is finite) and we get the extended notation: A/B/m/K

- queueing discipline,

where K is the capacity of the station (queue + server). This means that if the number of jobs at server and queue is K, a newly arriving job is lost.

212 6.2

SINGLE STATION QUElJElNG SYSTEMS

PERFORMANCE

MEASURES

The different types of queueing systems are analyzed mathematically to determine performance measures from the description of the system. Because a queueing model represents a dynamic system, the values of the performance measures vary with time. Normally, however, we are content with the results in the steady-state. The system is said to be in steady state when all transient behavior has ended, the system has settled down, and the values of the performance measures are independent of time. The system is then said to be in statistical equilibrium, i.e., the rate at which jobs enter the system is equal to the rate at which jobs leave the system. Such a system is also called a stable system. Transient solutions of simple queueing systems are available in closed-form, but for more general cases, we need to resort to Markov chain techniques as described in Chapter 5. Recall that the generation and the solution of large Markov chains can be automated via stochastic reward nets [MuTr92]. The most important performance measures are: Probability of the Number of Jobs in the System rk: It is often possible to describe the behavior of a queueing system by means of the probability vector of the number of jobs in the system rk. The mean values of most of the other interesting performance measures can be deduced from 7’rk: xk = P[there are Ic jobs in the system]. Utilization p: If the queueing system consists of a single server, then the utilization p is the fraction of the time in which the server is busy, i.e., occupied. In case there is no limit on the number of jobs in the single server queue, the server utilization is given by: P=

mean service time arrival rate X = I-* mean interarrival time service rate p

(6.3)

The utilization of a service station with multiple servers is the mean fraction of active servers. Since mp is the overall service rate:

p=X v’ and p can be used to formulate the condition for stationary mentioned previously. The condition for stability is:

PC 1,

(64 behavior

(6.5)

i.e., on average the number of jobs that arrive in a unit of time must be less than the number of jobs that can be processed. All the results given in Chapters 6-10 apply only to stable systems.

PERFORMANCE MEASURES

213

Throughput A: The throughput of an elementary queueing system is defined as the mean number of jobs whose processing is completed in a single unit of time, i.e., the departure rate. Since the departure rate is equal to the arrival rate X for a queueing system in statistical equilibrium, the throughput is given by:

in accordance with Eq. (6.4). W e note that in the case of finite buffer queueing system, throughput can be different from the external arrival rate. Response Time T: The response time, also known as the sojourn time, is the total time that a job spends in the queueing system. Waiting Time W: The waiting time is the time that a job spends in a queue waiting to be serviced. Therefore we have: Response time = waiting time + service time. Since W and T are usually random numbers, their mean is calculated. Then: (6.7)

The distribution functions of the waiting time, Fw(x), time, FT(~), are also sometimes required. Queue Length

and the response

Q: The queue length, Q, is the number of jobs in the queue.

Number of Jobs in the System K: The number of jobs in the queueing system is represented by K. Then:

k=l

The mean number of jobs in the queueing system K and the mean queue length Q can be calculated using one of the most important theorems of queueing theory, Little’s theorem:

K=AT, and

G = Xw.

Little’s theorem is valid for all queueing disciplines and arbitrary systems. The proof is given in [LittGl].

(6.9) (6.10) GI/G/m

214

6.3

SINGLE STATION QUEUEING SYSTEMS

THE M/M/l

SYSTEM

Recall that in this case, the arrival process is Poisson, the service times are exponentially distributed, and there is a single server. The system can be modeled as a birth-death process with birth rate (arrival rate) X and a constant death rate (service rate) CL.We assume that X < ,Qso the underlying CTMC is ergodic and hence the queueing system is stable. Then using Eq. (3.12)) we obtain the steady-state probability of the system being empty: 1 7ro

=

00

1 k-l =

1-tm-I;

1+

g(qky

k=l

k=li=O

p

which can be simplified to: 1

x1-x. I-L

?ro=l+*

From Eq. (3.11) we get for the steady-state probability jobs in the system: n,+ = TO

0

that there are k

k

-

k 2 0.

,

CL x

rk=(l-‘)

(

A’” )

/J-CL

or with the utilization

*

p = X/,X no-l-p

(6.11)

and: rk

=

(1 -

P)Pk,

(6.12)

the probability mass function (pmf) of the modified geometric random variable. In Fig. 6.2, we plot this pmf for p = l/2. The mean number of jobs is obtained using Eqs. (6.12) and (6.8):

x=21-p.

(6.13)

In Fig. 6.3, the mean number of jobs is plotted as a function of the utilization p. This is the typical behavior of all queueing systems. From Eq. (1.10) we obtain the variance of the number of jobs in the system: 2 -P

OK - (1 - p)2

(6.14)

THE M/M/l

SYSTEM

215

k

fig. 6.2 The solution for ~TTI, in a M/M/l

system.

and the coefficient of variation:

With Little’s theorem (Eqs. (6.9) and (6.10)) we get for the mean response time:

T=

lh l-p’

(6.15)

with Eq. (6.7) for the mean waiting time: (6.16) and with Little’s theorem again for the mean queue length:

&=p2 l-p’

(6.17)

The same formulae are valid for M/G/l-PS and M/G/l-LCFS preemptive resume (see [Lave83]). For M/M/l-FCFS we can get a relation for the response time distribution if we consider the response time as the sum of k + 1 independent exponentially distributed random variables [Tkiv82]:

x+x1+&+“‘+&,

216

SINGLE STATION QUEUEING SYSTEMS zo-

15--

K

lo--

5--

1

0.8

P Fig. 6.3

The mean number of jobs ?? in a M/M/l

system.

where X is the service time of the tagged job, Xr is the remaining service time of the job in service when the tagged job arrives, and X2,. . . , XI, are the service times of the jobs in the queue; each of these is exponentially distributed with parameter CL.Note that Xr is also exponentially distributed with rate p due to the memoryless property of the exponential distribution. Noting that the Laplace-Stieltjes transform (LST) of the exponentially distributed service time (see Table 1.5) is:

I-L Lx(s) = ___ p+s’ we get for the conditional LST of the response time:

Unconditioning using the steady-state probability the response time is:

Eq. (6.12), the LST of

LT(S) =co(&)Ic+l *(1-P)Pk LT(S)

=

I41

-

P)

s + PC1 -

P> *

(6.18)

THE M/M/m

Thus the response time 7’ is exponentially CL0 - 4:

distributed

SYSTEM

217

with the parameter

FT(x) = 1 -e -P(l--Pb

(6.19)

and the variance: var(T) = Similarly we get the distribution

1

of the waiting time: x = 0,

1 - P, ev(x)

(6.20)

P2(l - d2’

(6.21)

=

1 - p . e--cLC1-P12, x > 0. Thus Fw(O) = P(W = 0) = 1 - p is the mass at origin, corresponding to the probability that an arriving customer does not have to wait in the queue.

6.4

THE M/M/co

SYSTEM

In an M/M/co queueing system we have a Poisson arrival process with arrival rate X and an infinite number of servers with service rate ,LLeach. If there are k jobs in the system, then the overall service rate is kp because each arriving job immediately gets a server and does not have to wait. Once again, the underlying CTMC is a birth-death process. Prom Eq. (3.11) we obtain the steady-state probability of k jobs in the system:

with Eq. (3.12), we obtain the steady-state probability tem:

of no jobs in the sys-

1 A’“1 k=l

=

e-t,

0 i-J

‘Ic!

(6.22)

and finally:

(6.23)

218

SINGLE STATION QUEUEING SYSTEMS

This is the Poisson lmf, and the expected number of jobs in the system is:

K=

x.

(6.24)

P With Little’s theorem the mean response time as expected:

(6.25)

6.5

THE M/M/m

SYSTEM

An M/M/m queueing system with arrival rate X and service rate p for each server can also be modeled as a birth-death process with:

Jc2 0,

XI, = 4 O
b%

Pk =

i *t%

The condition for the queueing system to be stable (underlying CTMC to be ergodic) is X < mp. The steady-state probabilities are given by (from Eq. (3.11)):

rk=

k-l

m-l TO rI

rI i=m

With an individual server utilization,

rk

=

x *IQ’

p = A/(mp),

we obtain:

(6.26)

(

I

pkmm TO*! ’

JC2 *,

and from Eq. (3.12) -’

(6.27)

THE M/M/l/K

FINITE CAPACITY SYSTEM

219

that an arriving customer has to wait in the

The steady-state probability queue is given by:

Pm = P(K 2 m) = 5

rk

k=m

(6.28)

b-v)” = m!(l - p) * 7ro Using Eqs. (6.26) and (6.8) we obtain for the mean number of jobs in the system: E=mpfL-Pm,

(6.29)

1-P

and for the mean queue length: Q=P.p,. 1-P

(6.30)

From this the mean response time ?? and mean waiting time w by Little’s theorem (Eqs. (6.9) and (6.10)) can be easily derived. A formula for the distribution of the waiting time is given in [GrHa85]: &v(x)

= {

l-pm, 1 _ pm . e-mP(l-Pb,

x = 0, x > 0.

(6.31)

Figure 6.4 shows the interesting and important property of an M/M/m system that the mean number of jobs in the system K increases with the number of servers m if the server utilization is constant but the mean queue length s decreases.

6.6

THE M/M/l/K

FINITE CAPACITY

SYSTEM

In an M/M/l/K queueing system, the maximum number of jobs in the system is K, which implies a maximum queue length of K - 1. An arriving job enters the queue if it finds fewer than K jobs in the system and is lost otherwise. This behavior can be modeled by a birth-death process with: x = k

A, 09-K { 0,

/& =p

k 2 K,

k= l,...,K.

Using Eqs. (3.12), (3.11), and a = X/p, we obtain the steady-state probability of k jobs in the system: OLk K.

(6.32)

220

SINGLE STATION QUEUEING SYSTEMS

m Fig. 6.4 Mean queue length 8 and mean number of jobs in the system K as functions of the number of servers m.

Since there are no more than K customers in the system, the system is stable for all values of X and CL.That is, we need not assume that X < p for the system to be stable. In other words, since the CTMC is irreducible and finite, it is ergodic. If X = CL,then a = 1 and: 1 TO =

K+1

= TIC,

k = 1,2, . . . , K.

(6.33)

The mean number of jobs in the system is given by (using Eq. (6.8)): ,

~a l-a

K ,2’

-

K+l 1 - (JK+l

. $+I

7

a# 1, (6.34) a = 1.

Note that the utilization p = 1 - ~0 # X/p in this case; it is for this reason that we labeled the quantity a = X/,Q in the preceding equations. The throughput in this case is not equal to X, but is equal to X(1 - no). Similar results have been derived for an M/M/m/K system (see [Klei75], [AllegO], [GrHa85], or [Triv82]).

221

MACHINE REPAIRMAN MODEL

6.7

MACHINE

REPAIRMAN

MODEL

Another special case of the birth-death process is obtained when the birth rate Xj is of the form (M - j)X, j = 0, 1, . . . , A4, and the death rate, ,~j = ,Q. This structure can be useful in the modeling of interactive computer systems where a an individual terminal user issues a request at the rate X whenever it is in the “thinking state.” If j out of the total of m terminals are currently waiting for a response to a pending request, the effective request rate is then (A4 - j)X. The request completion rate is denoted by ,X Similarly, when M machines share a repair facility, the failure rate of each machine is X and the repair rate is ,Y (see Fig. 6.5). Since the underlying CTMC is irreducible and

Fig. 6.5

Machine repairman model.

finite, it is ergodic implying that the queueing system is always stable. The expressions for steady-state probabilities are obtained using Eqs. (3.12) and (3.11) as: rk = TQ

k-1 X(A4 - i) rI CL i=o

or: (6.35)

Hence: 7ro =

1 M

(6.36)

k x

w k=O

’

The utilization of the computer (or the repairman) is given by p = (1 - ~0) (see Eq. (6.11)) and the throughput by ~(1 - ~0). With the average thinking time l/X, we get for the mean response time of the computer: TX

AL? &-no)

-- 1 J-l

and, using Little’s theorem, the average number of jobs in the computer:

-I( = j’,,f _ id1 ; To).

(6.38)

222

6.8

SINGLE STATION QUEUEING SYSTEMS

CLOSED TANDEM

NETWORK

Consider the closed tandem network in Fig. 6.6 with a total number of K L

fig. 6.6

Closed tandem network.

customers. This network is known as the cyclic queueing network. Assume that node 1 has service rate p, and node 2 has service rate ~2. The state space in this case is {(ICI, Jcz) 1 Icr 2 0, Ic2 2 0, Icr + lc:! = K} where Ici (i = 1,2) is the number of jobs at node i. Such multidimensional state spaces, we modify the earlier notation for steady-state probabilities. The joint probability of the state (IQ, Ic2) will be denoted by r(Icr, Icz) while the marginal pmf at node i will be denoted by ni(lci). Although, the underlying CTMC appears to have a two-dimensional state space, in reality it is one-dimensional birth-death process because of the condition Icr + i& = K. Thus the marginal pmf of node 2 can be derived from Eqs. (3.12) and (3.11):

k2

r2(k2) = 0, k2 > K.

k2 I K;

9

1 r2(0)

= 1+5

i=l

(g

and with u = I_L~/,x~: fl2@2)

(6.39)

=

l-u 1 _ @+l ’

u# 1, (6.40)

1 K+l’

u = 1.

Similarly, we obtain for node 1:

n(h) =m(O) ---&, 1-i 1 v(O)

u# 1,

(i)““’

(6.42)

=

I The utilizations

(6.41)

1 K+l’

u= 1.

are given by: P11 -

Tl (O),

P2 =

1 -

7r2(0),

(6.43)

THE M/G/l

223

SYSTEM

and the throughput: x = Xl

= x2 = p1p1

(6.44)

= p2p2,

and the mean number of customers in node 2 with equation: 772 = Q(O)

-5

ii3

*uk2

k2=0

a - cK Uk2 = 7r2(0) *?A& k2=0 8

=

7r2(0) *ux

=--

U

l-u K1=K-K2.

*

1 -

ukz+l

l-u (K+l)*uK+l 1 -UK+1

uz1 ’

(6.45)

7

(6.46)

6.9 THE M/G/lSYSTEM The mean waiting time w of an arriving job in an M/G/l components:

system has two

1. The mean remaining service time we of the job in service (if any), 2. The sum of the mean service times of the jobs in the queue. We can sum these components to:

W=Wo+Q-TB,

(6.47)

where the mean remaining service time we is given by: we = P(server is busy) . z + P(server is idle) . 0,

(6.48)

with the main remaining service time x of a busy server (the remaining service time of an idle server is obviously zero). When a job arrives, the job in service needs R time units on the average to be finished. This quantity is also called the mean residual life and is given by [Klei75]: T; TB R = --$- = -+1+

c”,).

B

For an M/M/l

system (c”, = l), we obtain:

72M/M/l

=TB

=

;,

(6.49)

224

SINGLE STATION QUEUEING SYSTEMS

which is related to the memoryless property of the exponential distribution. The probability P(server is busy) is t,he utilization p by definition. From Eq. (6.47) and Little’s theorem s = X . w, we obtain:

wo WE--1-P

(6.50)

and with Eq. (6.49) finally: 2

(1+

n=&*

2

4)

'

(6.51)

the well-known Pollaczek-Khintchine formula (see Eq. (3.28)) for the mean queue length. For exponentially distributed service time we have ci = 1 and for deterministic service time we have c& = 0. Thus: (see Eq. (6.17)),

The mean queue length is divided by two if we have deterministic service time instead of exponentially distributed service times. Using the theory of embedded DTMC (see Section 3.2.1), the Pollaczek-Khintchine transform equation can be derived (Eq. (3.25)):

G@) = B-(X - AZ) (’ - p)(l - ‘1 B”(X - AZ) - x’

(6.52)

where G(x) is the z-transform of steady-state probabilities of the number of jobs in the system rk and B”(s) is the LST of the service time. As an example we consider the M/M/l system with B” (s) = p/ (s + p) (see Table 1.5). From Eq. (6.52) it follows that:

(+) = l-p

1 - pz

G(x) = F(l

- p)pk - 2’.

k=O

Hence we find the steady-state probability

of Ic jobs in the system:

TI, = (1 - P)Pk.

THE GI/M/l

SYSTEM

225

With Eqs. (6.52) and (1.57) we obtain for the mean number in the system [Klei75] (see also Eq. (3.28)):

1+c; &p+p2.l-p 2

(6.53)

or, using Eq. (6.7) and Little’s theorem:

K=p$ifJ.

(6.54)

Hence:

&-

p2 1-P

y3,

which is the same as Eq. (6.51). Figure 6.7a shows that for an M/G/l system the mean number of jobs K increases linearly with increasing squared coefficient of variation, and that the rate of increase is very sensitive to the utilization. From Fig. 6.7b we see how dramatically the mean number of jobs K increases if the utilization approaches 1 and that this is especially the case if the squared coefficient of variation cg is large.

6.10 THE GI/M/lSYSTEM We state the results for GI/M/l

systems, using the parameter o given by:

CT= A-(/L - /LO) where A”(s) is the LST of the interarrival mean number of jobs in the system: j&-L-

(6.55)

time [Tane95]. For example, the

(6.56)

1-O’

the variance of the number of jobs in the system: p(l+*

2

OK=

(l-42

- P) ’

(6.57)

the mean response time: T,L.-

1 /!A l-a’

(6.58)

Pea 1-O’

(6.59)

the mean queue length: Q=

226

SINGLE STATION QUEUEING SYSTEMS

= 0.9

= 0.8

= 0.7 = 0.5

=3

=

1

= 0

Fig. 6.7 (a) Mean number of jobs ?? in an M/G/l system as a function of & with parameter p. (b) M ean number of jobs I? in a M/G/l system as a function of p with parameter ci .

THE GI/M/l

SYSTEM

227

the variance of the queue length:

0; = PO + 4 - P>> (l-a)2 ’

(6.60)

the mean waiting time:

and the waiting time distribution: &v(x)

= { 1 - 0, x = 0, 1 _ 0. e-P(l--a)z 7 x > 0.

(6.63)

In the case of an M/M/l system, we have A-(s) = X/(s + X) (see Table 1.5). Together with Eq. (6.55) we obtain:

and with the Eqs. (6.56)-(6.63), the well-known M/M/l formulae. A more interesting example is the Ez/M/l system. From Table 1.5 we have: 2

A-(s)= ( & > ) and with Eq. (6.55) we have

Using Eqs. (6.56)-(6.63), we obtain explicit formulae for Ez/M/l performance measures. system is very similar, espeThe behavior of an M/G/l and of a GI/M/l cially if c$ S-1. This is shown in Fig. 6.8 where we compare the mean number of jobs K for M/G/l and GI/M/l systems having the same coefficient of variation cg. Note that c$ in the case of GI/M/l denotes the coefficient of variation of the interarrival times, while in the M/G/l case it denotes the -coefficient of variation of the service times. Note also that in the M/G/l case, K depends only on the first two moments of the service time distribution, while in the GI/M/l case, the dependence is more extensive. In Fig. 6.8, for the GI/M/l system, the gamma distribution is used. For increasing value of c$ the deviation increases.

228

SINGLE STATION QUEUEING SYSTEMS c”x = 10/i

p

= 4

20

t

GI/M/l __-______ M/G/l

=

1

= 0.1

0:2

Fig. 6.8

6.11

0:4

0:3

”

0:5 ‘.’ P

0.6

:‘:I’:““‘: 0:7

Mean number of jobs K in an M/G/l

THE GI/M/m

0:8

and a GI/M/l

019

system.

SYSTEM

Exact results for GI/M/m queueing systems are also available, See [Alle90], [Klei75], or [GrHa85]. The mean waiting time is given by: jYj%

J m/!L(1 - 0)2 ’

(6.64)

where: a=A”(mp--mpo)

(6.65)

and: J= For Rk, see [Klei75] (page 408). We introduce only a heavy traffic approximation: (6.66) which is an upper bound [Klei75] and can be used for GI/G/l and GI/G/m system as well (see subsequent sections). Here X is the reciprocal of the mean interarrival time, ai is the variance of interarrival time, and a$ is the variance of service time.

THE G/,/G/l SYSTEM

229

6.12 THE GI/G/lSYSTEM In the GI/G/l case only approximation formulae and bounds exist. We can use M/G/l and GI/M/l results as upper or lower bounds, depending on the value of the coefficient of variation (see Table 6.1). Table 6.1 Upper bounds (UB) and lower bounds (LB) for the GI/G/l mean waiting time

4

43

>l >l
>l l
M/W

GI/M/l

LB LB UB UB

LB UB LB UB

Another upper bound is given by Eq. (6.66) with m = 1 [Klei75]: (6.67)

A modification of this upper bound is [Marc78]: (6.68)

This formula is exact for M/G/l and is a good approximation for GI/M/l and GI/G/l systems if p is not too small and ci or ci are not too big. A lower bound is also known [Marc78]: w

> P2 * c2B + P(P - 2)

2X(1-p)

'

(6.69)

but more complex and better lower bounds are given in [Klei76]. Many approximation formulae for the mean waiting time are mentioned in the literature. Four of them that are either very simple and straightforward or good approximations are introduced here. First, the well-known AllenCunneen approximation formula for GI/G/m systems is [AllegO]: (6.70)

This formula is exact for M/G/l (Pollaczek-Khintchine formula) and a fair approximation elsewhere and is the basis for many other better approximations. A very good approximation is the Kramer/Langenbach-Belz formula, a direct extension of Eq. (6.70) via a correction factor:

(6.71)

230

SINGLE STATION QUEUEING SYSTEMS

with the correction factor: exP --*-* 3 G KLB

P ( 2 1-P

(1-c?J2 ) 05 cA5 19 c; + ci >

=

(6.72)

CA > 1.

(=P

Another extension of the Allen-Cunneen formula is the approximation Kulbatzki [Kulb89]:

of

(6.73) with:

f 1,

CA E {%I},

[p(14.1cA - 5.9) + (-13.7cA + 4.1)] c; + [p(-59.7ca I + [&A

+ 21.1) + (54&A - 16.3)] cg

- 4.5) + (-1.5cA + 6.55)],

o
; -0.75~ + 2.775,

CA > 1,

(6.74 It is interesting to note that Eq. (6.74) was obtained using simulation experiments. A good approximation for the case ci < 1 is the Kimura approximation [Kimu85]:

c2,+c;wz -wM/M/rn 2

6.13

THE M/G/m

((I-&)exp(v)

+&)-l.

(6.75)

SYSTEM

We obtain the Martin’s approximation formula for M/G/m systems [Mart721 by an extension of the Eq. (6.47) for the mean waiting time for an M/G/l system: w=w

0 +”

m

**T

(6.76)

Because of the m servers, an arriving customer has to wait, on the average, only for the service of Q/m customers. The remaining service time in this case is: Tvo=Pm*R.

(6.77)

THE M/G/m SYSTEM

231

With Eq. (6.49) we obtain as an approximation: (6.78) and with Eqs. (6.76) and (6.77) finally: w

c

pmlP

. C1 +43)

2m

l-p

(6.79)

*

This is a special case of the Allen-Cunneen formula for GI/G/m systems (see next section) and is exact for M/M/m and M/G/l systems. For the waiting probability Pm we can use Eq. (6.28) for M/M/m systems or a simple yet good approximation from [Bolc83]:

(6.80)

As an example, we compare the exact waiting probability mation for m = 5 in Table 6.2.

with this approxi-

Tab/e 6.2 Exact (Eq. (6.28)) and approximate (Eq. (6.80)) values of the probability of waiting P

0.2

0.4

0.6

0.7

0.8

0.9

0.95

0.99

Pme, PmwP

0 0

0.06 0.06

0.23 0.21

0.38 0.34

0.55 0.56

0.76 0.75

0.88 0.86

0.97 0.97

From Fig. 6.9 we see that the deviation for the mean number of jobs K in the system is very small if we use the approximation for Pm instead of the exact value. A good approximation for the mean waiting time in M/G/m systems is due to Cosmetatos [Cosm76]: &c/m

% CB 2w

M/M/m

+ (I-

In Eq. (6.81), we can use Eq. (6.30) for FM/M/m WM/D/m

=

1 2

-

(6.81)

cf?)~M/D/m*

and use:

1 ’ nCDm

’ WM/M/m

where: ncDm =

I t (1- p)(m- 1)&TY5ii-2 16pm

-l >

’

(6.82)

232

SINGLE STATION QUEUEING SYSTEMS

--

20

= 10

= 5 = 2

Fig. 6.9 Mean number of jobs in an M/M/m formulae for P,.

system with exact and approximative

For WM,Dlrn we can also use the Crommelin approximation formula [Crom34]: (ypdUrn

(vm)! (6.83)

Boxma, Cohen, and Huffels [BCH79] a1so use the preceding formulae for w M/D/m as a basis for their approximation: (6.84)

wiV/G/m

where:

a=

1 (‘i+l)71 i 1, m- 1 --

m+l

(

and: 1 - c; 71 X-+---$

m+l

G?

1 , m m>l= 1,7

233

THE GI/G/m SYSTEM

Tijms [Tijm86] uses yi from the BCH-formula tion:

Fv,G,m= ((1

-

P)Ylm

+ $(c:,

(Eq. (6.84)) in his approxima-

+ 1))

(6.85)

WIv,M,m.

------mm_ DES 8--

ac: Allen-Cunneen cro: Crommelin co: Cosmetatos tij: Tijms bch: Boxma, Cohen,

tij bch

Huffles

l--

4 0

Fig. 6.10

Comparison of different M/G/5

approximations with DES (p = 0.7).

In Fig. 6.10 we compare five approximations introduced above with the results obtained from discrete-event simulation (DES). From the figure we see that for p = 0.7 all approximations are good for ci < 2, and that for higher values of ci the approximation due to Cosmetatos is very good and the others are fair.

6.14

THE GI/G/m

SYSTEM

For GI/G/m systems only bounds and approximation formulae are available. These are extensions of M/G/m or GI/G/l formulae. We begin with the well-known upper bound due to Kingman [King’i’O]: (6.86) and the lower bound of Brumelle [Brum71] and Marchal [Marc74]:

- PC2 - P>--.-m-l wz P2& m Nl - P>

cg+l (6.87)

2P

-

234

SINGLE STATION QUEUEING SYSTEMS

As a heavy traffic approximation duced for GI/M/m systems:

we have Eq. (6.66), which we already intro-

(6.88) and the Kingman-Kiillerstrijm distribution:

approximation

Fw(x) M 1 - exp

(

-

[Kii1174] for the waiting time

W-P) 1 0; +ai/m2

(6.89)

’ Xx > *

The most known approximation formula for GI/G/m systems is the AllenCunneen (A-C) formula [AllegO]. We already introduced it for the special case GI/G/l (Eq. (6.70)). Note that the A-C formula is an extension of Martin’s formula (Eq. (6.79)) w h ere we replace the 1 in the term (1 + c”,) by ci to consider approximately the influence of the distribution of interarrival times:

KnlP c; + c2B was:.l-p

2m

(6.90)

*

For the probability of waiting we can either use Eq. (6.28) or the good approximation provided by Eq. (6.80). As in the GI/G/l case, the AllenCunneen approximation was improved by Kramer/Langenbach-Belz [KrLa76] using a correction factor: Pm//L c; + c2 *B * GKLB, l-p 2m

w&z -

exp GKLB

=

21-P(1-4)2 ) ()
9

cl + c; >

exP

and by Kulbatzki CA in Eq. (6.90):

(6.91)

(6.92)

CA > 1,

[Kulb89] using the exponent f (CA, cB, p) in place of 2 for p,lp

$A

,CB4) + cg

w=:. 1-P

2m

*

(6.93)

For the definition of f(cA, cg, p), see Eq. (6.74). The Kulbatzki formula was further improved by Jaekel [JaekSl]. We start with the Kulbatzki GI/G/lformula and use a heuristic correction factor to consider the number of servers m:

(6.94)

235

THE G//G/m SYSTEM

This formula is applicable even if the values of m and the coefficients of variation are large. In order to extend the Cosmetatos approximation [Cosm76] from M/G/m to GI/G/m systems, FM/M/m and WM,D,~ need to be replaced by WGI,M,~ and m~r,n,~, respectively:

w GI/G/m where

2wGI/M/m

!-z CB

+ (l

-

C$i)~GI/D/rn,

is given by Eq. (6.64) or by the following approximation:

WGI/M/m

$cZ, + l)exp (-i.

A$.

‘:i$“)

m WGI/M/m

forO
5 I,

A

x

for CA > 1,

CA’CB “) + 1 WM,M,~ >

(6.96) and

WGI/D/m

by:

wGI/D/m

=

1 2

1

(6.97)

’ WGI/M/m> ncDm

- * -

with ncDm from Eq. (6.82) or: WGIIDIm z Cy,m)f(cA,o@) . jqYMIDlm (6.98) h(p, m) = 4J(m

- l)/(m

+ 4) f (1 - p) + 1

with WM,Dirn from Eq. (6.82) or Eq. (6.83). A good approximation case ci 2 l’is given by Kimura [Kimu85]:

-1

1 - ci 1 - 4c(m, p) exp c(m, p) = (1 - P>(m - 1)

for the

+

dG-2 16pm

l- 43 l+c(m,p)

+c2

+ c; - 1 A >

7

’ (6.99)

Finally the Boxma, Cohen, and Huffels formula GI/G/m systems: 2WGI/D/m WGI/G/m

=

2 I(’

+ “)

2&&I,D,m

[BCH79] can be extended to

w

+ (1 -:($;I,M,

(6.100) m

236

SINGLE STATION QUEUEING SYSTEMS

(b)

sac

Fig. 6.11 (a) Mean queue length g of a GI/G/lO 6 of a GI/G/m system with p = 0.8.

system, and (b) mean queue length

237

PRIORITY QUEUEING

1 =

0.95

= 0.9

= 0.7

Fig. 6.12

Mean queue length of a GI/G/m

as well as the Tijms

system with r-n = 1,5,10.

formula [Tijm86]:

~GI/G/m=

((1

-

P)Ylm

+

$c;

+

1,)

~GI,M/rn,

(6.101)

with a and yi from Eq. (6.84). In Figs. 6.11 and 6.12 the mean queue lengths as functions of the coefficients of variation ci, ci, the utilization p, and the number of servers m are shown in a very compact manner using the approximate formula of Allen-Cunneen. Fig. 6.13 shows that the approximations of Allen-Cunneen, Kramer/Langenbath-Belz, and Kulbatzki are close together, especially for higher values of the coefficients of variation c$ with a large number of servers m.

6.15

PRIORITY

QUEUEING

In a priority queueing system we assume that an arriving customer belongs to a priority class T (T = 1,2,. . . , R). The next customer to be served is the customer with the highest priority number r. Inside a priority class the queueing discipline is FCFS.

238

SINGLE STATION QUEUEING SYSTEMS

=

1

= 0

I

O'O.6

0.65

0.7

0.75

0.8

0.85

019

P

Fig. 6.13 (a) Mean queue length 0 for a GI/G/lO system with cl = 0.5. (b) Mean queue length Q for a GI/G/lO system with cl = 2.0.

239

PRIORITY QUEUEING

6.15.1

System

without

Preemption

Here we consider the case where a customer already in service is not preempted by an arriving customer with higher priority. The mean waiting time w, of an arriving customer of priority class T has three components: 1. The mean remaining service time we of the job in service (if any). 2. Mean service time of customers in the queue that are served before the tagged customer. These are the customers in the queue of the same and higher priority as the tagged customer. 3. Mean service time of customers that arrive at the system while the tagged customer is in the queue and are served before him. These are customers with higher priority than the tagged customers. Define: Ni,:

Mean number of customers of class i found in the queue by the tagged (priority r) customer and receiving service before him,

a+:

Mean number of customers of class i who arrive during the waiting time of the tagged customer and receive service before him.

Then the mean waiting time of class r customers can be written as the sum of three components: (6.102) For multiple server systems (m > 1): R N. 1 w,=w,+~“T-+~m.-, i=l

RYiTi,

m GUI i=l

1 Pi

(6.103)

where Ni, and YVi, are given by: 7Vir = 0 i < r, (6.104) and, with Little’s theorem: T&. = xgi

i 2 r, (6.105)

We can solve Eqs. (6.102) and (6.103) to obtain: TV,=

WO

(1 - G)(l

-

(6.106) Or+l)

240

SINGLE STATION QUEUEING SYSTEMS

where: R 0,

=

(6.107)

pi*

c i=r

The overall mean waiting time is:

R xi w=cx.wi.

(6.108)

i=l Values for wo are given by Eqs. (6.48), (6.49), (6.77), and (6.78) and are the weighted sum of the woi of all the classes: R WO,M/M/l

(6.109)

= Cpid7 i=l R

WO,M/G/l

=

C i=l

pi

1+ * 2

c;

(6.110)

2t%

’

R

(6.111) R WO,GI/G/l,KLB

R5 C i=l R

WO,GI/G/l,KUL

C3 c i=l

c$

+ c&

(6.112)

* GKLB,

Pi ’

=ZPi f(CA

z lcB,

pi ’ cAz

di)

+

3-G

cii

7

(6.113)

(6.114)

(6.115)

(6.116) cl, mO,GI/G/m,KLB

=

-

+ cg. ’ GKLB,

(6.117)

Pi

(6.118) see Eq. (6.74); for GKLB,GI/G/l, see Eq. (6.72); and for see Eq. (6.92).

FOT ~(cA~, cgi, pi), GKLB,GI/G/~,

PRIORITY QUEUEING

Also, the GI/G/m-FCFS to priority queues:

Approximation

241

of Cosmetatos can be extended

with: WGI/M/m,

WO,GI/M/m

= (l-

a,)(1

-G-+1)’

WO,GI/D/m &,D,mr

= (1

WO,GI/D/m

Fig. 6.14

e

-

G4l

PTnCR pi

2mP

i=l

Mean queue lengths g’, for an M/M/l

-

G-+1)

’

$Ai,O,Pi)

*

’ Pi

priority system without preemption.

The M/G/m Cosmetatos approximation can similarly be extended. All GI/G/m approximations yield good results for the M/G/m-priority queues as well. Figure 6.14 shows the mean queue length G, for different priority classes of a priority system without preemption and R = 3 priority classes together

242

SINGLE STATION QUEUEING SYSTEMS

with the mean queue length for the same system without priorities. It can be seen that the higher-priority jobs have a much lower queue length, and the lower-priority jobs have a much higher mean queue length than the FCFS system. 6.15.2

Conservation

Laws

In priority queueing systems, the mean waiting time of jobs is dependent on their priority class. It is relatively short for jobs with high priority and considerably longer for jobs with low priority because there exists a fundamental conservation law ([Klei65, Schr70, HeSo841): R c i=l

pJvi

=

pwo

=

p * WFCFS

(6.120)

1-P

or: PiTTi = 1-P

FCFS-

(6.121)

The conservation law to apply the following restrictions must be satisfied: 1. No service facility is idle as long as there are jobs in the queue, i.e., scheduling is work conserving [ReKo75]. 2. No job leaves the system before its service is completed. 3. The distributions of the interarrival times and the service times are arbitrary with the restriction that the first moments of both the distributions and the second moment of the service time distribution exist. 4. The service times of the jobs are independent of the queueing discipline. 5. Preemption is allowed only when all jobs have the same exponential service time distribution and preemption is of the type preemptive resume. 6. For GI/G/ m systems all classes have the same service times.

restriction is not necessary for GI/G/l

This

systems [HeSo84].

If for GI/G/ m systems the service times of the classes differ, then the conservation law is an approximation [Jaek9 11. 6.15.3

System

with

Preemption

Here we consider the case that an arriving customer can preempt a customer of lower priority from service. The preempted customer will be continued later at the point where he was preempted (preemptive resume). Thus a customer

PRIORITY QUEUEING

243

- 1. Hence of class T is not influenced by customers of the classes 1,2,...,r we need to consider only a system with the classes T, r + 1 9- - * , R to calculate

wr

=

-r - WO

(6.122)

1 -Or7

where wb is the mean remaining service time for such a system and we obtain it by substitution of p by or in the formulae. For example: P;

-r

W OAC

=

2ma,

-

R c

c; + cg pi

(6.123)

i=r

We sum only from r and R rather than from 1 to R. For P&, we replace p by or in the exact or approximative formula for Pm. To obtain the mean waiting time of a customer of class r we apply the conservation law twice:

i=r

(6.124) -r+l

Or+l’W

R

= c

pi ‘Wis

i=r+l

By substitution

we obtain:

W, = A- (OrWr- CTr+l~r+l)*

(6.125)

Pr

Equation (6.125) is exact for M/,&I/l and M/G/l systems. It is also exact for M/M/m systems if the mean service time of all classes are the same. For other systems, Eq. (6.125) is a good approximation. From Fig. 6.15 it can be seen that even for heavy utilization the mean queue length for the highest priority class is negligible in the case of preemption, at the expense of poorer service for lower-priority classes. 6.15.4

System

with

Time-Dependent

Priorities

Customers with low priorities have long waiting times especially when the utilization of the system is very high. Sometimes it is advantageous for a customer priority to increase with the time, so that he does not have to wait too long. This possibility can be considered if we do not have a fixed priority function: ar(t) = Priority

of class r at time t.

244

SING1 E STATION QUEUEING SYSTEMS

=

2

= 3

Fig. 6.15 emption.

Mean queue lengths for an M/M/l

priority

system with and without pre-

Such systems are more flexible but need more expense for the administration. A popular and easy to handle priority function is the following:

%@)= (t - to) *b,

(6.126)

(see Fig. 6.16a), with 0 < br 5 b2 5 . . . 5 b,. The customer enters the system at time to and then increases the priority at the rate b,. Customers of the same priority class have the same rate of increase b, but different values of the arrival time to. We only consider systems without preemption and provide the following recursive formula [BoBr84] for the mean waiting time of priority class-r customer:

TV, =

(6.127)

with the same mean remaining service times wa as for the priority systems with fixed priorities. If these are exact, then Eq. (6.127) is also exact; otherwise it is an approximation formula. Another important priority function is:

q&) = r’, + t - to,

(6.128)

PRIORITY QUEUEING

(4

245

Priority A

Fig. 6.16 (a) Priority function with slope b,. (b) Mean queuelength 0, for an M/M/l priority system with time-dependent priorities having slope b,.

with 0 5 ~1 5 ~2 5 . . . 5 T,. Each priority class has a different starting priority rr and a constant slope (see Fig. 6.17a). In [BoBr84] a heavy traffic approximation (as p approaches 1) is given: (6.129) as well as a more accurate recursive formula: (6.130) Note that for m = 1, we have P, = p. From Figs. 6.16b and 6.17b it can be seen that systems with time-dependent priorities are more flexible than systems with static priorities. If the values of

246

SINGLE STATION QUEUEING SYSTEMS

b, or rT, respectively, are close together, then the systems behave like FCFS systems, and if they are very different, then they behave more like systems with static priorities. In many real time systems, a job has to be serviced within an upper time limit. To achieve such a behavior, it is advantageous to use a priority function that increases from 0 to 00 between the arrival time and the upper time limit ul-. A convenient priority function is:

G-(t) =

(4

(t-to)&-t+to) { co

to cKu,+to, UT+ to 5 t.

(6.131)

Priority c

rr i to

4

t”0

-t =l = 5

= 10

Fig. 6.17 (a) Priority function with starting priority Tr. (b) Mean queue lengths &, for an M/M/l priority system with time-dependent priorities having starting priorities rr.

PRIORITY QUEUEING

to 4

tb t:, + G, GP Gi

247

to+&b

(b)

WV

Fig. 6.18 (a) P riority function with upper time limit ur. (b) Mean waiting time w, for an M/M/l priority system with static priorities and with upper time limits ur = 2,4,6.

Again we get a heavy traffic

approximation

(as p approaches

1):

(6.132)

248

SINGLE STATION QUEUEING SYSTEMS

and a recursive formula for more accurate results: TV, x

6 --fJ-FVi(l-~) 1-P (

(I-Pmexp(--C2)))

-(l-iglpi(l-z)

(l-P.,,~exp(-!Zk)))~l.

(6’133)

Also see [BoBr84] and [Jaekgl] for other priority functions. In Fig. 6.18b, the mean waiting times for a system with static priorities are compared to those of a priority system with upper time limits. The figure also contains the upper time limits. For the two higher priorities, the static priority system is better, but the upper time limit system is better over all priorities.

6.16

THE

ASYMMETRIC

SYSTEM

The calculation of the performance measures is fairly simple if there is only one server or several servers with identical service times arranged in parallel, and the interarrival time and service time are exponentially distributed (M/M/m queueing systems). However, heterogeneous multiple servers often occur in practice and here the servers have different service times. This situation occurs, for example, when machines of different age or manufacturer are running in parallel. It is therefore useful to be able to calculate the performance measures of such systems. One method for solving this problem is given in [BoScSl] in which the formulae derived provide results whose relative deviation from values obtained by simulation are quite acceptable. The heterogeneous multiple server is treated in more detail by [BFH88] and [Baer85] who achieve exact results. Also see [Triv82] and [GeTr83] for exact results. We will assume throughout this section that the arrival process is Poisson with rate X and that service times are exponentially distributed with rate ,Q; on the ith server. 6.16.1

Approximate

Analysis

In homogeneous (symmetric) queueing systems, the response time depends only on whether an arriving job finds some free server. A heterogeneous (asymmetric) multiple server system differs since which of the m servers processes the job is now important. The analysis in [BoScSl] is based on the following consideration: Let p,+ (k = 1, . . . , m) be the probability that a job is assigned to the &h server. Because it can be assumed that a faster server processes more jobs than a slower server, it follows that the ratios of the probabilities p!, to each other are the same as the ratios of the service rates to each other. Here no distinction is made between different strategies for selecting a

THE ASYMMETRIC SYSTEM

249

free server. This approximation is justified by the fact that in a system with heavy traffic there is usually one free server and therefore a server selection strategy is not needed. Then pk (1 5 /C5 m) is approximately given by: Pk

&!L-

(6.134) Et%’ i=l

In addition, pk can be used to determine the utilization

pk of the lath server: (6.135)

i=l

i=l

Thus the utilization is given by:

of all the servers is the same and the overall utilization (6.136)

p = pk.

In [BoScSl] it is shown that the known formulae for symmetric M/M/m and GI/G/m systems can be applied to asymmetric M/M/m and GI/G/m systems if Eqs. (6.135) and (6.136) are used to calculate the utilization p. Good approximations to the performance measures are obtained provided the values of the service rates do not differ too much, i.e, pmin/pmax 5 10 (see Table 6.3).

Table 6.3 Approximative queueing systems

x (4

(b)

and simulative

results

for (a) M/M/5

and (b) M/E2/5

P@ >

WV

a%)

7x92)

0.799 0.807 0.808 0.826 0.807 0.801 0.799

0.819 0.812 0.844 0.858 0.839 0.830 0.824

2.472 2.472 2.472 2.472 2.472 2.472 2.472

2.367 2.476 2.443 2.549 2.456 2.442 2.425

2.495 2.500 2.626 2.713 2.606 2.569 2.523

0.800

0.825

1.854

1.854

1.962

I-lk

P(A)

PW

73

73 73 81.11 81.11 81.11 81.11

10,15,20,20,25 16,17,18,19,20 5,10,18,22,35 4,8,16,32,40 8,9,20,31,32 8,14,20,26,32 10,15,20,25,30

0.811 0.811 0.811 0.811 0.811 0.811 0.811

81.11

10,15,20,25,30

0.811

>

Notes: A = approximation, S1 = discrete-event simulation with random selection of a free server, and 5’2 = discrete-event simulation with selection of the fastest free server.

6.16.2

Exact

Analysis

This section describes the methods used for calculating the characteristics of asymmetric queueing systems described in [BFH88] and [Baer85]. These systems are treated as a CTMC, and the performance measures are calculated

250

SINGLE STATION QUEUEING SYSTEMS

by solving the underlying CTMCs. In this way exact results are obtained for elementary asymmetric M/M/m queueing systems. This method is used as follows: First, the steady-state probabilities are calculated for systems without queues, known as loss systems. Then the formulae given in [BFH88] and [Baer85] can be used to extend the results to asymmetric, non-lossy systems. Two strategies for selecting a free server are investigated: 1. The job is assigned randomly to a free server as in [BFH88].

2. The fastest free server is selected as in [Baer85]. 6.16.2.1 Analysis of M/M/m 1 oss Systems A loss system contains no queues. The name indicates that jobs that cannot find a free server are rejected and disappear from the system. A CTMC is constructed and solved to obtain a closed-form formula for its steady-state probabilities. It should be noted that asymmetric queueing systems differ from symmetric queueing systems in that it is not sufficient when producing the CTMC to describe the states by the number of servers occupied. Due to the different service rates, the state classification depends on which of the m servers are currently occupied. Thus each state is characterized by a set g = (ICI, Ic2, . . . , k,), where k, belongs to g if the Zth server is currently occupied. Here i = (g( is the number of servers currently occupied (1 5 i < m) and I is an index (1 5 I 5 m).

Fig. 6.19 CTMC for an asymmetric M/M/3 free server.

An M/M/3 system is used as an example. diagram of the loss system is shown in Fig. 6.19. The

Random Selection of a free Server

The state transition

loss system with random selection of a

THE ASYMMETRIC SYSTEM

251

steady-state probabilities are given by [BFHM]:

9-7r - (m -‘g’Y rIx m!

*kEgG

(6.137)

*=”

for all g C G with g # 8 and G = {1,2,. . . , m}. In order to determine ;TT~,we use the normalization condition:

c

7rg = 1

with

G = {1,2,. . . ,m).

(6.138)

gEG The probability

of loss r CL)is the probability dL) m = r{l

I2., ._,m}

that all servers are occupied:

(6.139)

= TG.

The utilization pk of the kth server is equal to the sum of the state probabilities of all states in which the kth server is occupied:

d = c =g*

(6.140)

g: kEg

Choice of the Fastest Free Server. Here it is always assumed that the individual servers are sorted in descending order of the service rates, i.e., the fastest server has the lowest index. Let pm = (~1, ~2, . . . , bum) and let (pk-‘, pm) = of loss n!$) = ~~l,..,,ml(~m) is given (P1,P2,*.* ~~k-l,~m)* The probability by [Baer85]:

7T(L) m = ~{l,.,.,m}(Pm) = Bm(Pm) = Bm-l(pm-l)

&(CL”-‘,/km)

*

Bk(pkwl,pk

+ b7L>

(6.141)

-’

1

’

with: w4

The utilization pk =

= “iT{l}(Pl)

=

&.

pk of the individual servers is given by:

-$Bk-l(pk-‘)

-

Bk(pk)]

with

Bo(pO) = 1.

(6.142)

6.16.2.2 Extending to Non-Lossy System According to [BFH88], the next step is to use the results obtained so far to draw conclusions about the behavior of the corresponding non-lossy system. Here the met hod used to select a free server of the corresponding loss system does not matter. It is only necessary

252

SINGLE STATION QUEUEING SYSTEMS

that the values appropriate to the selected strategy are used in the formulae finally obtained. In what follows, a superscript W means that the variable is a characteristic of a non-lossy system; a superscript L means that the variable is a characteristic of the corresponding loss system. The steady-state probabilities are given by [BFH88]: m-i . ,w m

for

i > m,

(6.143)

with:

,wm = -7&y

(6.144)

N’

where: 7&y

N=l+-

l-c

*c

x

c=-

with

f? k=l

(6.145) pk’

It is now possible to calculate all the state probabilities of the asymmetric non-lossy system and all interesting performance measures. It should be noted that c is not the same as the utilization p. The method for calculating p is described in the following. The probability Pm that an arriving job is queued is equal to the sum of the probabilities of all states in which there are more than m jobs in the system: (6.146) i=m

Analogous to the symmetric case, the mean queue length 8 is given by:

Q= 2

(i-m).rjW)=E.

(6.147)

-

i=m+l

The mean waiting time can be calculated using Little’s theorem:

Q TV=x.

(6.148)

To calculate the utilization p it is first necessary to calculate the specific utilization pk for each server: + 7&y * c/( 1 - c) N

.

(6.149)

i=m+l

Then p is given by: (6.150)

THE ASYMMETRIC SYSTEM

6.16.3

Exact

Analysis

of an Asymmetric

M/M/2

253

System

In [Triv82] an exact solution is given for an asymmetric M/M/2 system with the strategy of selecting the fastest free server. Figure 6.20 shows the CTMC state transition diagram for this system. The state of the system is defined

Fig. 6.20

CTMC for the M/M/2

asymmetric system.

to be the tuple (ICI, k 2) w h ere ICI 2 0 denotes the number of jobs in the queue including any at the faster server, and k2 E (0, 1) denotes the number of jobs at the slower server. If we use Eq. (2.58) for the underlying CTMC, we can calculate the steadystate probabilities easily with: c=

x

-

(6.151)

tQ1-k P2

r(k) 1) = cr(k - 1,1) = c’“%(l,

l),

for

k > 1.

(6.152)

Furthermore: n(O,l) = 5 %o, 1+ 2c p2

o>,

l+c x 7T(l,O) = --7q4 1+2cp1

o),

n(l,l)

= -

c

1+2c

Finally, using the normalization n-(0,0) = [l +

x(x+Il”)T(o Plj.42

(6.153)

70)*

condition, we get: X(X + Y2>

p&(1

+ 2c)(l - J1*

(6.154)

And with these, the utilizations of the two servers: p1 = 1 - T(O,O) - 7T(O,l), p2 = 1 - 7T(O,O)- 7+,0),

(6.155)

254

SINGLE STATION QUEUEING SYSTEMS

and the mean number of jobs in system: 1 A(1 - c)~’

j&

(6.156)

where: +-

’

l-c

1 ’

Example 6.1 To show how to use the introduced formulae for asymmetric M/M/m systems, we apply them to a simple M/M/2 example with X = 0.2, ~1 = 0.5, and ~2 = 0.25. Approximation

With Eq. (6.136) for the utilization:

[BoSc91]

0.2 = 0.267, 0.5 + 0.25

p’p1’p2’x= CL1+ P2

with Eq. (6.29) for symmetric M/M/2 -

K=2p+-r

systems for mean number in system: 2P3

1 - p2

0.575,

and with Little’s theorem (Eq. (6.9)):

T = 2.875. Exact [Triv82]

With Eq. (6.151): c = 0.267,

with Eqs. (6.153), (6.154), and (6.155) for the utilizations

O.a(O.2+ 0.25) 0.25 a0.5(1 + 2c)(l - c)

of the servers:

-I

1

= 0.6096,

7r(O,1) =

0.267

- 0.2 +0.6056 = 0.085, 1 + 0.534 0.25

7r(l,O) = -1.267 . -0.2 . 0.6056 = 0.2014, 1.534 0.5 l-0.6096 $0.085 = 0.305, p2 = 1 - 0.6096 + 0.2014 = 0.189,

p1 =

THE ASYMMETRIC SYSTEM

255

and with Eq. (6.156): A = 0.5 - 0.25(1 + 0.534) + 1 = 3.495, 0.2(0.2 + 0.25) 0.733 1 = 0.533, 3.495 + 0.7332

K=

K T = x = 2.663. Exact [BFH88] First we consider the asymmetric M/M/2 loss system with random selection of a free server: With Eq. (6.156) for the state probabilities: (2-l)! 5TT1=-. 2! (2-l)! 7r2 = - 2!

x 1 0.2 7ro= 0.2X(), j-jy’*o=y~’ x 1 ‘/12’ro=y~’

(2 -‘I! -.-.-.

7r1,2 =

2!

A

A

Pl

P2

ro

0.2

=

7r()= 0.47r(),

lfjT

0

*

O*

With the normalizing condition: (1 + 0.2 + 0.4 + 0.16)no = 1, Now the probability

x0 = 0.568.

of loss can be determined (Eq. (6.139)): ?p

and also the utilizations

= 7rl,2 = 0.16.0.568 = 0.091, of the servers (Eq. (6.140)):

pf = n1 + 7r1,2= (0.2 + 1.6) - 0.568 = 0.204, pi = r2 + r1,2 = (0.4 + 1.6) e0.568 = 0.319. This result means that the slower server has a higher utilization as expected. Now we can use these results of the loss system to obtain performance measures for the non-lossy system. For c and N we get with Eq. (6.145): ’ =

0.2 = 0.267, 0.5 + 0.25

and for the probability

N=l+

0.091 . 0.267 = 1 033 0.733 LY

of waiting (Eq. (6.146)): 1 0.091 p2=---.= 0.120. 0.733 1.033

The mean queue length and the mean waiting time (Eqs. (6.147), (6.148)): Q=

0.120.0.267 = 0.0437, 0.733

w = 0.219.

256

SINGLE STATION QUEUEING SYSTEMS

Utilization

of the servers (Eq. (6.149)): (w) _ 0.204 + 0.091 .0.267/0.733 = o 230 b? 1.033

Pl

(w) _ 0.319 + 0.091 s0.267/0.733 = o 341 L. 1.033

P2

Utilization

of the system (Eq. (6.150)): P=

2

= 0.275.

I-+&i 0.230

.

Mean number of jobs in system: E = PI + ~2

+ 62 = 0.230 + 0.341 + 0.0437 = 0.6147.

Mean response time: K T = x = 3.074. Exact [Baer85] The last case we consider is the asymmetric M/M/2 system with selection of the fastest free server using the formulae in [Baer85]. First we again consider the loss system and determine the probability of loss (Eq. (6.141)): r2

CL) = B2(p2) = B&2)

=-. x x+/-Q

1+ E.

Bl(P2)

x

R(Pl-tP2)

[

1+!% x---1

1 -l

-1

x x:‘“2 I

x+cL1+I.L2

0.2 + 0.25 + 0.5 -’ 0.2 + 0.25

1

= 0.0785, and the utilizations of the servers (Eq. (6.142)): pf = +30(po) Pi = g

[&W)

- B&d)]

= g

-B2(P2)]

= g

[l-

1

0.2Oe2 + 0.5 = 0.286,

.

[020;20

1 A*

5 - 0.0785 = 0 166

Again we use the results of the loss system to obtain performance measures for the queueing system with selection of the fastest server. From Eq. (6.145)): c = 0.267,

N=l+

0.0785 mO.267= l 0286 +* 0.733

THE ASYMMETRIC SYSTEM

Probability

257

of waiting (Eq. (6.146)): 1 0.0785 p,=-.-= 0.733 1.0280

0.104.

Mean queue length (Eq. (6.147)): 0.104 ’ 0.267 = 0.0379. 0.733 Mean waiting time: WC

Utilizations

g = L0 190 x of the servers (Eq. (6.149)): py

=

0.286 + 0.0785 e0.267/0.733 = o 306 L, 1.0286

(w) _ 0.166 + 0.0785 .0.267/0.733 1.0286

P2

Utilization

=

0.189.

of the system (Eq. (6.150)): 2 P=

1

= 0.234.

0.306 +1 0.189

Mean number of jobs in the system: x

= ~1 + p2

+ s = 0.306 + 0.189 + 0.0379 = 0.533.

Mean system time: K T = x = 2.665. In Table 6.4, results of this example are summarized.

In Table 6.5, the

Table 6.4 Results for several performance measures with different solution techniques and strategies for a asymmetric M/M/Z system (A = 0.2, ~1 = 0.5, ~2 = 0.25) Strategy Reference

Awr . [BoScS l]

FFS [Triv82]

FFS [Baer85]

Random [BFH88]

P Pl P2 E T

0.267 0.267 0.267 0.575 2.875

(0.267) 0.305 0.189 0.533 2.663

0.234 0.306 0.189 0.533 2.665

0.275 0.230 0.341 0.615 3.074

Notes: FFS = selection of the fastest free server and Random = random selection of the free server.

mean response time T for an asymmetric M/M/2

system is compared with

258

SINGLE STATION QUEUEING SYSTEMS

-Table 6.5 Mean response times T, TI, and Ta for an asymmetric M/M/2 system with strategy FFS and two M/M/l systems (A = 0.2, ~2 = 0.25, ~1 = cv e~2) a

Tl 572

T

1

2

3

4

5

20 20 4.762

3.33 20 2.662

1.818 20 1.875

1.25 20 1.459

0.55 20 1.20

the mean response times Ti (for a M/M/l system with service rate pi) and ??z (for a M/M/l system with service rate ,CQ)[Triv82]. From Table 6.5 we see that sometimes it is better to disconnect the slower server if we wish to minimize the mean response time. This is the case only for low utilization of the servers and if the service rates differ considerably. This issue is further explored in [GeTr83].

6.17

SYSTEMS WITH

BATCH SERVICE

An interesting and important way to service customers in queueing systems is the batch service. In such batch systems the service of all the customers of a batch is started at the same time. There are many applications of this kind of service, especially in manufacturing systems. Two policies are in use. In the case of a full batch policy (FB), the service of the batch is started when all b customers of the batch have arrived. In the case of the minimum batch policy (MB), a minimum number of a customers is sufficient to start the service of the batch. If there are more then b waiting customers, only b customers are collected to a batch and serviced together. The extended Kendall notation for batch systems is: GI/G[“~b]/m, M u It’iserver system with MB policy GI/G[blb]/m, M u It’iserver system with FB policy A special case of the MB policy is the GREEDY policy where the service begins when a = 1 customer is in the queue. To obtain a formula for the mean waiting time and the mean queue length for the GI/G[b~bl/m queueing system, we use the already introduced formulae for GI/G/m queueing systems and calculate the mean queue length $ for the batches. From the result we can calculate the mean queue length abatch for the individual customers. If the arrival rate of the individual customers is X, then the arrival rate of the batches with batch size b is:

x A= b,

(6.157)

SYSTEMS WITH BATCH SERVICE

and the coefficient of variation for the batch interarrival

259

time is: (6.158)

using Eq. (1.52) with the coefficient of variation of the individual customers CA. Using the queue length 0 of the batches, the queue length of the individual customers is approximately given by [HaSp95, KirsSl]:

The first part of the formula is due to the customers in the dj batches of the queue, and the second part is the mean number of customers who are still not combined to a batch. The accuracy of Eq. (6.159) depends mainly on the accuracy of the applied approximation formula for calculating 0. A solution for GI/GI”>bl/ m queueing systems is given by a heuristic extension of Eq. (6.159) [KirsSI]: -b,bl

Q

batch

wbdj+P,

(6.160)

Eq. (6.160) includes Eq. (6.159) if a = b. For the probability of waiting Pm, we use the corresponding value for the M/M/m queueing system. In the special case of an M/M[b~61/l system, we obtain an Eb/M/l queueing system for the calculation of 0 since in this case the interarrival time of the batch is the sum of b exponentially distributed interarrival times. In the case of an M/MIb7’]/m system, we obtain an Eb/M/m queueing system for which exact formulae (see Eqs. (6.59) and (6.64) respectively) are known. From Fig. 6.21~ it can be seen how the mean queue length Qbatch for individual jobs depends on the batch size b in a full batch system. Figure 6.21b shows that for increasing values of p, the mean queue length of a minimum batch system approaches that of a full batch system with the same batch size b. Figure 6.22 shows that the queue length Qbatch of an individual job in a minimal batch system depends more on the batch size b than on the minimum batch size a, as also can be seen from Eq. (6.160). Moreover we realize that a has a noticeable influence only on Gk$Lh for small values of p. Problem 6.1 Consider two M/M/l systems with arrival rate Xi = 0.5/ set and service rate ~1 = l/set each and an M/M/2 system with arrival rate x2 = 2x1 = l/ set and service rate ,LQ= ,~i = l/set for each server. Compare utilization pi, mean number of jobs in the system ?7i, mean waiting time wi, and mean response time Fi for both cases (i = 1,2). Discuss the results! Problem 6.2 Consider an M/M/l/K system and determine the throughput and the mean number of jobs K as a function of the maximum number

260

SINGLE STATION QUEUEING SYSTEMS

c_____- -----

0-0 .1

012

0:3

Ol4

0:5 P

016

017

o:a

019

Fig. 6.21 (u) M ean queue length Gbatch for a full batch system M/M[‘?b]/l-FCFS and (b) ?#$~~ for a minimum batch system.

SYSTEMS WITH BATCH SERVICE

261

p = 0.8

p = 0.6 p = 0.3

Fig. 6.22

Mean queue length QL&

of a minimum batch system.

of jobs in the system K for the cases X = ,x/2, X = p, and X = 2,~ with ,Q= l/ sec. Discuss the results. Problem 6.3 Compare the mean number of jobs K of an M/D/l, an M/M/l, and an M/G/l system (cg2 - 2) with X = 0.8/set and p = l/set. Give a comprehensible explanation why the values are different although the arrival rate X and the service rate h are always the same. Problem 6.4 Compare the mean number of jobs E( of an M/G/l and a GI/M/l system with X = O.g/sec, p = l/set, ci = 0.5, and cg = 0.5, respectively. Discuss the results. Problem 6.5 Consider a GI/G/l system with p = 0.8, cl = 0.5, and c; = 2. Compute the mean queue length s using the Allen-Cunneen, the Also compute KrRmer/Langenbach-Belz, and the Kimura approximation. upper and lower bounds for the mean queue length (see Section 6.12). Problem 6.6 Consider an M/M/l priority system with 2 priority classes, and ~1 = ~2 = l/set and Xr = X2 = 0.4/ sec. Compute the mean waiting time w for the two classes for a system with and without preemption. Discuss the results and compare them to the corresponding M/M/l-FCFS system. Problem

6.7

Consider an asymmetric M/M/2

system with X = l/set,

~1 - l/set, and ~2 - l.S/sec. Cjompute the utilizations p1 and p2 of the two servers and the mean number of jobs r using the approximation

method and

262

SINGLE STATION QUEUEING SYSTEMS

the exact methods for the strategy random selection of the free server and the strategy selection of the fastest free server (see Section 6.16). Discuss the results. Problem 6.8 Consider an IvI/M[~J~]/~ batch system and an M/M[275]/1 batch system with p = 0.5. Compute the mean queue length 8 for both cases and discuss the results.

7 Queueing Networks

Queueing networks consisting of several service stations are more suitable for representing the structure of many systems with a large number of resources than models consisting of a single service station. In a queueing network at least two service stations are connected to each other. A station, i.e., a node, in the network represents a resource in the real system. Jobs in principle can be transferred between any two nodes of the network; in particular, a job can be directly returned to the node it has just left. A queueing network is called open when jobs can enter the network from outside and jobs can also leave the network. Jobs can arrive from outside the network at every node and depart from the network from any node. A queueing network is said to be closed when jobs can neither enter nor leave the network. The number of jobs in a closed network is constant. A network in which a new job enters whenever a job leaves the system can be considered as a closed one. In Fig. 7.1 an open queueing network model of a simple computer system is shown. An example of a closed queueing network model is shown in Fig. 7.2. This is the central-server model, a particular closed network that has been proposed by [Buze71] for the investigation of the behavior of multiprogramming system with a fixed degree of multiprogramming. The node with service rate ,ur is the central-server representing the central processing unit (CPU). The other nodes model the peripheral devices: disk drives, printers, magnetic tape units, etc. The number of jobs in this closed model is equal to the degree of multiprogramming. A closed tandem queueing network with two nodes is shown in Fig. 7.3. A very frequently occurring queueing network is the machine repairman model, shown in Fig. 7.4. 263

264

QUEUEING NETWORKS

s&yvb;f

Disk

L,,o

Ii: CPU

Printer

cl -‘T-p Stream of completed Magnetic Tape jobs

fig. 7.1 Computer system shown as an open queueing network. I

Fig, 7.2

Central-server model.

There are many systems that can be represented by the machine repairman model, for example, a simple terminal system where the M machines represent the M terminals and the repairman represents the computer. Another example is a system in which A4 machines operate independently of each other and are repaired by a single repairman if they fail. The M machines are modeled by a delay server or an infinite server node. A job does not have to wait; it is immediately accepted by one of the servers. When a machine fails, it sends a repair request to the repair facility. Depending on the service discipline, this request may have to wait until other requests have been dealt with. Since there are M machines, there can be at most .M repair requests.

fig. 7.3 Closed tandem network.

DEFINITIONS AND NOTATION

265

Fig. 7.4 Machine repairman model.

This is a closed two-node queueing network with h/i7jobs that are either being processed, waiting, or are in the working machines.

7.1

DEFINITIONS

AND NOTATION

We will consider both single class and multiclass networks. 7.1.1

Single Class Networks

The following symbols are used in the description of queueing networks: N

Number of nodes

K

The constant number of jobs in a closed network

(h,

k2,

. . .,lc~) The state of the network

k

The number of jobs at the ith node; for closed networks

mi

zki=K i=l The number of parallel servers at the ith node (mi > 1)

Pi

Service rate of the jobs at the ith node

lIPi

The mean service time of the jobs at the ith node

Pij

Routing probability, the probability that a job is transferred to the jth node after service completion at the ith node. (In open networks, the node with index 0 represents the external world to the network.)

POj

The probability that a job entering the network from outside first enters the jth node

Pi0

The probability

that a job leaves the network just after com-

pleting service at node i (pi0 = 1 - 5 pij). j=l XOi

The arrival rate of jobs from outside to the ith node

266

x

QUEUEING NETWORKS

The overall arrival rate from outside to an open network N

(A= c Xoi) i=l

the overall arrival rate of jobs at the ith node The arrival rate Xi for node i = 1,. . . , N of by adding the arrival rate from outside and the nodes. Note that in statistical equilibrium the is equal to the rate of arrival, and the overall written as:

an open network is calculated arrival rates from all the other rate of departure from a node arrival rate at node i can be

N

Xi = Xoi + C Xjpji 7 for i = 1, . . . , N

(74

j=l

for an open network. These are known as traffic equations. For closed networks these equations reduce to: Xi = c

Xjpji,

for i = 1,. . . , N,

(74

j=l

since no jobs enter the network from outside. Another important network parameter is the mean number of visits (ei) of a job to the ith node, also known as the visit ratio or relative arrival rate:

xi ei = -, x

for i=

l,...,N,

(7.3)

where X is the overall throughput of the network(see Eqs. (7.24) and (7.25)). The visit ratios can also be calculated directly from the routing probabilities using Eqs. (7.3) and (7.1), or (7.3) and (7.2). For open networks, since Xoi = X*pOi: N

ei =po; + Cejpji, j=l

for i = l,...,

N,

(7.4)

and for closed networks:

(7.5) Since there are only (N - 1) independent equations for the visit ratios in closed networks, the ei can only be determined up to a multiplicative constant. Usually we assume that er = 1, although other possibilities are used as well. Using ei, we can also compute the relative utilixation xi, which is given by:

(74

DEFINITIONS AND NOTATION

267

It is easy to show that [ChLa74] the ratio of server utilizations is given by:

7.1.2

Multiclass

Networks

The model type discussed in the previous section can be extended by including multiple job classes in the network. The job classes can differ in their service times and in their routing probabilities. It is also possible that a job changes its class when it moves from one node to another. If no jobs of a particular class enter or leave the network, i.e., the number of jobs of this class is constant, then the job class is said to be closed. A job class that is not closed is said to be open. If a queueing network contains both open and closed classes, then it is said to be a mixed network. Figure 7.5 shows a mixed network. The following additional symbols are needed to describe queueing networks that contain multiple job classes, namely: R

The number of job classes in a network

k.ar

The number of jobs of the rth class at the ith node; for a closed network: N

R

cc Kr

kir = K

(74

The number of jobs of the rth class in the network; not necessarily constant, even in a closed network: N

kir = Kr

c

i=l

(7.9)

K

The number of jobs in the various classes, known as the population vector (K = (KI, . . . , KR))

Si

The state of the ith node (Si = (ki,, . . . , kiR)): N c

Si = K

(7.10)

i=l

S

The overall state of the network with multiple classes (S = (S

Pir

The service rate of the ith node for jobs of the rth class

Pir,js

that a job of the rth class at the ith node is transferred to the sth class and the jth node (routing probability)

po,js The probability in an open network that a job from outside the network enters the jth node as a job of the sth class

268

QlJEUElNG NETWORKS

The probability in an open network that a job of the rth the network after having been serviced at the ith node, so:

Pir,o

Pir,O = 1 - 5

2

j=l

x

(7.11)

Pir,js

s=l

The overall arrival rate from outside to an open network

Xa,ir The arrival rate from outside to node i for class r jobs (Xe,ir = X. PO,+) A.al-

The arrival rate of jobs of the rth class at the ith node: (7.12) j=l

s=l

for closed networks, pg,i, = 0 (1 < i < N, 1 < r < R) and we obtain:

&r = 5 j=l

5

(7.13)

xjs * Pjs,ir.

s=l

The mean number of visits e;, of a job of the rth class at the ith node of an open network can be determined from the routing probabilities similarly to Eq. (7.4):

eir = PO,ir + 2 j=l

2

ejsPjs,i7-,

for i=l,,..,

N, r=l,...,

R.

(7.14)

R.

(7.15)

s=l

For closed networks, the corresponding equation is: for i=l,...,

N, r=l,...,

Usually we assume that er, = 1, for r = 1, . . . , R, although other settings are also possible.

7.2 7.2.1

PERFORMANCE

MEASURES

Single Class Networks

Analytic methods to calculate state probabilities and other performance measures of queueing networks are described in the following sections. The determination of the steady-stateprobabilities r(kl,. . . , k~) of all possible states of the network can be regarded as the central problem of queueing theory. The

PERFORMANCE MEASURES

269

mean values of all other important performance measures of the network can be calculated from these. There are, however, simpler methods for calculating these characteristics directly without using these probabilities. Note that we use a slightly different notation compared with that used in Chapters 2-5. In Chapters 2-5, xi(t) denoted the transient probability of the CTMC being in state i at time t and 7ri as the steady-state probability in state i. Since we now deal with multidimension state spaces, ~(ki, Icz, . . . , IAN) will denote the steady-state probability of state (ki, lc2, . . . , k~). The most important performance measures for queueing networks are: Marginal Probabilities ri(lc): For closed queueing networks, the marginal probabilities xi(k) that the ith node contains exactly ki = k jobs are calculated as follows: 7ri(k) =

c 5

~(kl,...,kN).

(7.16)

k,=K

j=1

& ki=k

Thus xi(k) is the sum of the probabilities of all possible states (kl, . . . , 5 kj = K where a fixed j=l number of jobs, k, is specified for the ith node. The normalization condition that the sum of the probabilities of all possible states (ICI, . . . , kN> kN), 0 5 ki 5 K that satisfy the condition

that satisfy the condition

5 kj = K with

(0 5 kj 5 K)

must be 1,

j=l

that is: c

7r(kl,...,kN)

= 1.

(7.17)

Correspondingly, for open networks we have:

ri(k)= c r(kl,...Jw), k,=k

with the normalization

condition:

C(OTTICI,...

,kN)=L

Now we can use the marginal probabilities to obtain other interesting performance measures for open and closed networks. For closed networks we have to take into consideration that xi(k) = 0 Vk > K. Utilization by:

pi: The utilization pi of a single server node with index i is given

pi

=

ITi

k=l

(7.18)

270

QUEUNNG NETWORKS

where pi is the probability

that the ith node is busy, that is: pi = 1 - 7ri(O).

(7.19)

For nodes with multiple servers we have: pi = k

Fmin(mi, 2 k=O

k);lri(lc) = 1 - mgl G.,,(i,), k=O

(7.20)

and if the service rate is independent of the load we get (see Eq. (6.4)): pi zz - Ai mini ’

(7.21)

Throughput Xi: The throughput Xi of an individual node with index i represents in general the rate at which jobs leave the node: (7.22) where the service rate pi(Jc) is, in general, dependent on the load, i.e., on the number of jobs at the node. For example, a node with multiple servers (m; > 1) can be regarded as a single server whose service rate depends on the load pi (Ic) = min(k,mi). pi, where ,!~i is the service rate of an individual server. It is also true for load-independent service rates that (see Eqs. (6.4), and (6.5)):

We note that for a node in equilibrium, arrival rate and throughput are equal. Also note that when we consider nodes with finite buffers, arriving customers can be lost when the buffer is full. In this case, node throughput will be less than the arrival rate to the node. Overall Throughput A: The overall throughput X of an open network is defined as the rate at which jobs leave the network. For a network in equilibrium this departure rate is equal to the rate at which jobs enter the network, that is:

x = 2 xoi.

(7.24)

i=l

The overall throughput of a closed network is defined as the throughput of a particular node with index i for which ei = 1. Then the overall throughput of jobs in closed networks is: (7.25)

PERFORMANCE MEASURES

Mean Number given by:

of Jobs xi:

271

The mean number of jobs at the ith node is

Ki = &i(k).

(7.26)

k=l

From Little’s theorem (see Eq. (6.9)) it follows:

Ki = A&

(7.27)

where Fi denotes the mean response time. Mean Queue Length mined by:

&i: The mean queue length at the ith node is deter-

(7.28) or, using Little’s theorem:

6.i = XiWi,

(7.29)

where Wi is the mean waiting time. Mean Response Time Ti: The mean response time of jobs at the ith node can be calculated using Little’s theorem (see Eq. (6.9)) for a given mean number of jobs f;;i :

Ti=r.

Ki i

(7.30)

Mean Waiting Time wi: If the service rates are independent of the load, then the mean waiting time at node i is: wigi-l. Pi

7.2.2

Multiclass

(7.31)

Networks

The extension of queueing networks to include multiple job classes leads to a corresponding extension of the performance measures. The state probability of a network with multiple job classes is represented by r(Sr, . . . , S,). The normalization condition that the sum of the probabilities of all possible states (Sl,.. . , S,) is 1 must also be satisfied here.

272

QUEUEING NETWORKS

----

Source

-

Open Class Closed Class

7

Fig. 7.5

A mixed network.

Marginal Probability n;(k): For closed networks the marginal probability, i.e., the probability that the ith node is in the state Si = k, is given by:

n(k)= c +h,..vSv), -$

(7.32)

S,=K

3=1

&

S,=k

and for open networks: 7ri(k) = c

(7.33)

x(S1,. . . , S,).

s,=k

The following formulae for computing the performance measures can be applied to open and closed networks.

Utilization pir: The utilization of the ith node with respect to jobs of the rth class is: Pir --

?ri(k)?

c

-f, ’

all states

min(rni, Ici),

Ici = f

a

k

&,

(7.34)

r=l

with k, > 0

and if the service rates are independent on the load: Air Pir

(7.35)

= m&7-

Throughput Air: The throughput Air is the rate at which jobs of the rth class are serviced and leave the ith node [BrBa80] : (7.36) all states k with k, > 0

2

PRODUCT-FORM QUEUEING NETWORKS

273

or if the service rates are independent on the load: Ai, = mi.pirspir.

(7.37)

Overall Throughput A,: The overall throughput of jobs of the rth class in closed networks with multiclasses is:

+xiT,

(7.38)

ei,

and for open networks:

k-= 5 xo,ir.

(7.39)

i=l

Mean Number of Jobs %?i,: The mean number of jobs of the rth class at the ith node is: Ic,. q(k).

(7.40)

all states k with Ic,. > 0

Little’s theorem can also be used here: Ki, = Ai,* Fir.

(7.41)

Mean Queue Length gir: The mean queue length of class r jobs at the ith node can be calculated using Little’s theorem as:

Qir = Xi,wip*

(7.42)

Mean Response Time Ti,: The mean response time of jobs of the rth class at the ith node can also be determined using Little’s theorem (see Eq. (7.41)): (7.43) Mean Waiting Time Wi,: If the service rates are load-independent, the mean waiting time is given by:

wi, ZTi, - ‘. t%r 7.3

PRODUCT-FORM

QUEUEING

then

(7.44)

NETWORKS

In the rest of this chapter, we will discuss queueing networks that have a special structure such that their solutions can be obtained without generating their underlying state space. Such networks are known as product-form or separable networks.

274

QUEUEING NETWORKS

7.3.1

Global

Balance

The behavior of many queueing system models can be described using CTMCs. A CTMC is characterized by the transition rates between the states of the corresponding model (for more details on CTMC see Chapters 2-5). If the CTMC is ergodic, then a unique steady-state probability vector independent of the initial probability vector exists. The system of equations to determine the steady-state probability vector 7r is given by 7rQ = 0 (see Eq. (2.58)), where Q is the infinitesimal generator matrix of the CTMC. This equation says that for each state of a queueing network in equilibrium, the flux out of a state is equal to the flux into that state. This conservation of flow in the steady state can be written as: c j&s

rjqji

= 7ri

c

qij,

‘d&S,

(7.45)

jES

where qij is the transition rate from state i to state j. After subtracting ni eqii from both sides of Eq. (7.45) and noting that qii = - Cjfi qij, we obtain the global balance equation (see Eq. (2.61)): Vi E S :

c i#i

r’jqji

- 7ri

c

4ij = 0,

(7.46)

i#i

which corresponds to the matrix equation rrQ = 0 (Eq. (2.58)). In the following we use two simple examples to show how to write the global balance equations and use them to obtain performance measures. Example 7.1 Consider the closed queueing network given in Fig. 7.6. The network consists of two nodes (N = 2) and three jobs (K = 3). The service times are exponentially distributed with mean values l/pi = 5 set and l/p2 = 2.5 set, respectively. The service discipline at each node is FCFS. The state space of the CTMC consists of the following four states: ((3, O), c&l>, (1, a>, (0,3)}*

Fig. 7.6

A closed network.

The notation (ki, /&) says that there are /cl jobs at node 1 and k2 jobs at node 2, and r(lcl, Ic2) denotes the probability for that state in equilibrium. Consider state (1,2). A transition from state (1,2) to state (0,3) takes place if the job at node 1 completes service (with corresponding rate ~1). Therefore, ,LL~is the transition rate from state (1,2) to state (0,3) and, similarly, ~2 is

PRODUCT-FORM QUEUEING NETWORKS

Fig. 7.7

State transition

diagram

for Example

275

7.1.

the transition rate from state (1,2) to state (2,l). The flux into a state of the model is just given by all arcs into the corresponding state, and the flux out of that state is determined from the set of all outgoing arcs from the state. The corresponding state transition diagram is shown in Fig. 7.7. The global balance equations for this example are: 439

oh1

= 7-42, qp2,

427

w/Q1

+ P2> =

+

2)t/% + P2) = 7@, ~)CLZ=

7r(3,0)/% + 7r(l, 2)/L& 7+, r(

l)/%

+ T(o,

3),Q,,

1,Qh.

Rewriting this system of equations in the form rrQ = 0 we have:

and the steady-state probability vector n = (7r(3,0), ~(2, l), n(l,2), ~(0,3)). Once the steady-state probabilities are known, all other performance measures and marginal probabilities xi(k) can be computed. If we use ~1 = 0.2 and ~2 = 0.4, then th e generator matrix Q has the following values: -0.2

‘;0

0.2

iof0

0

0

$26 0.4

-0.4 oo2 *

Using one of the steady-state solution methods introduced in Chapter 3, the steady-state probabilities are computed to be: 7r(3,0) = 0.5333,

7r(2,1) = 0.2667,

n&2)

= 0.1333,

~(0,3) = 0.0667

and are used to determine all other performance measures of the network, as follows: l

Marginal probabilities

(see Eq. (7.16)):

q(O) = x2(3) = 7r(0,3) = 0.0667,

q(l)

= n,(2) = +,2)

= 0.133,

276

QUEUEING NETWORKS

~~(2) = ~(1) l

Utilizations

= 7r(2,1) = 0.2667,

(see Eq. (7.20)):

p1 =

a Throughput

7rr(3) = 7r2(0) = n(3,O) = 0.5333.

1 -

7rl(O)

=

0.9333,

p2 = 1 - 7r2(0) = 0.4667.

(see Eq. (7.23)): x = h = x2 = plpl = pap2 = 0.1867.

l

Mean number of jobs (see Eq. (7.26)): El = 2

knl (k) = 2.2667,

X2 = 2

k=l

l

h2(k)

= 0.7333.

k=l

Mean response time of the jobs (see Eq. (7.30)): %

Xl

= ~1

= 12.1429,

T2

x2

= xz

= 3.9286.

Example 7.2 Now we modify the closed network of Example 7.1 so that the service time at node 1 is Erlang-2 distributed, while the service time at node 2 remains exponentially distributed. The modified network is shown in Fig. 7.8. Service rates of the different phases at node 1 are given by ~11 = ~12 = 0.4sec-l and the service rate of node 2 is ~2 = 0.4. There are K = 2 jobs in the sysiem. A state (ICI, I; Icz), I = 0, 1,2, . . ., of the network is now

Fig. 7.8 Network with Erlang-2 distributed server.

not only given by the number Ici of jobs at the nodes, but also by the phase 1 in which a job is being served at node 1. This state definition leads to exactly five states in the CTMC underlying the network. Pll

CL12

Pll

Pl2

Fig. 7.9 State transition diagram for Example 7.2.

PRODUCT-FORM QUEUEING NETWORKS

277

The state transition diagram of the CTMC is shown in Fig. 7.9. The following global balance equations can be derived: 451;

Oh1

= n(L

4272;

oh12

= 74271;

qp11

+ n(l,

al;

Wll

+ pa)

= 742,2;

O)p12

+ n(O,O; 2)/&,

492;

l)(Pl2

+ P2)

= +,1;

q/w,

2)/92

= +2;

l)p12.

7440;

1; qp2, 2; l)p2,

The generator matrix is:

-(I-h1

0 0

0 0

0 Pl2

Pll

+ cL2)

0

-(I412

0

IQ2

pY2

+p2)

*

-P2

After inserting the values ~11 = p12 = p2 = 0.4, Q becomes: -0.4

0.4 -0.4

0 0.4 -0.8

0

0

0.4

0 0 0.4 -0.8 0

0 0 0 0.4 -0.4

Solving the system of equations 7rQ = 0, we get: x(2,1; 0) = 0.2219,

n(2,2; 0) = 0.3336,

7r(l, 1; 1) = 0.2219,

7r(l,2; 1) = 0.1102,

r(O,O; 2) = 0.1125. These state probabilities are used to determine the marginal probabilities: q(O) = 7r2(2) = ?r(O,O;2) = 0.1125, ~~(1) = ~~(1) = x(1,1; 1) + ~(1,2; 1) = 0.3321, 7rr(2) = n2(0) = n(2,l; 0) + 7r(2,2; 0) = 0.5555. The computation of other performance measures is done in the same way as in Example 7.1. 7.3.2

Local

Balance

Numerical techniques based on the solution of the global balance equations (seeEq. (2.61)) can in principle alwaysbe used,but for large networksthis technique is very expensive because the number of equations can be extremely

278

QlJEUElNG NETWORKS

large. For such large networks, we therefore look for alternative solution techniques. In this chapter we show that efficient and exact solution algorithms exist for a large class of queueing networks. These algorithms avoid the generation and solution of global balance equations. If all nodes of the network fulfill certain assumptions concerning the distributions of the interarrival and service times and the queueing discipline, then it is possible to derive local balance equations, which describe the system behavior in an unambiguous way. These local balance equations allow an essential simplification with respect to the global balance equations because each equation can be split into a number of single equations, each one related to each individual node. Queueing networks that have an unambiguous solution of the local balance equations are called product-form networks. The steady-state solution to such networks’ state probabilities consist of multiplicative factors, each factor relating to a single node. Before introducing the different solution methods for product-form networks, we explain the local balance concept in more detail. This concept is the theoretical basis for the applicability of analysis methods. Consider global balance equations (Eq. (2.61)) for a CTMC: vi

E s :

c jES

Tiqii

= 7Ti

c j&3

4ij

or: rQ=O, with the normalization

condition: c

iTTi = 1.

iES

Chandy [Chan72] noticed that under certain conditions the global balance equations can be split into simpler equations, known as local balance equations. Local balance property for a node means: The departure rate from a state of the queueing network due to the departure of a job from node i equals the arrival rate to this state due to an arrival of a job to this node. This can also be extended to queueing networks with several job classes in the following way: The departure rate from a state of the queueing network due to the departure of a class r-job from node i equals the arrival rate to this state due to an arrival of a class r-job to this node. In the case of non-exponentially distributed service times, arrivals and departures to phases,instead of nodes, have to be considered.

PRODUCT-FORM QUEUEING NETWORKS

279

Fig. 7.10 A closed queueing network.

Example 7.3 Consider a closed queueing network (see Fig. 7.10) consisting of N = 3 nodes with exponentially distributed service times and the following service rates: ,LJ~= 4sec-‘, ~2 = lsec-l, and ,93 = 2sec-l. There are K = 2 jobs in the network and the routing probabilities are given as: p12 = 0.4, p13 = 0.6, and p21 = p31 = 1. The following states are possible in the network: (2,0, O),

((4% O),

(O,O, 3,

(171, o>, (LO,117(07 L0

The state diagram of the underlying CTMC is shown in Fig. 7.11. We set the overall flux into a state equal to the overall flux out of the state for each state to get global balance equations:

(1)

42,0,

WcLlP12

+ I-LlPl3)

= n(l,

0, qp3p31

(a>

407

27 Q2P21

= r(l,

1, QlP12,

(3)

40?

07 %-43P31

= +,

0, QJlP13,

+ YlP12)

= @,

2, qp2p21

(4)

417

17 Ob2P21

+ /JlP13

+ 40,1, c5)

+,

0,1)(~3P31

+ plPl2

+ plP13)

= +, + 64

(6)

~(0~

19 WcL3P31

+ P2P21)

= +,

+ r(l,

l,O)p2p21,

+ 42,0,0)/4p12

1b3P317

0,2)jJ3p31

+ T(O, 1, +2p21

O,O)/aP13, 1, qp1p13

+ n(l,

0, qp1p12.

To determine the local balance equations we start, for example, with the state (1, 1,O). The departure rate out of this state, because of the departure of a job from node 2, is given by ~(1,1,0)+,~2~p2r. This rate is equated to the arrival rate into this state (I, 1,O) due to the arrival of a job at node 2; 7r(2,0,0). ,~l.p12. Therefore, we get the local balance equation:

(4’)

+,1,0>*

ruZ’P21

= 7@,

60).

/Ql'Pl2.

Correspondingly, the departure rate of a serviced job at node 1 from state (1, 1,O) equals the arrival rate of a job, arriving at node 1 into state (l,l,O): (4”)

al,

0). Pl.

(P13 + P12) = n(0,

1, 1). /J3’P31 + n(O,2,0).

/Q.p21.

280

QUEUEING NETWORKS

Fig. 7.11 The CTMC for Example 7.3.

By adding these two local balance equations, (4’) and (4”)) we get the global balance Eq. (4). Furthermore, we see that the global balance Eqs. (l), (2), and (3) are also local balance equations at the same time. The rest of the local balance equations are given by: c5’)

T(l,

o,lh

(Pl2 417

(5”)

+ P13)

07 G3P31

= ‘@, = 7+,0,

1, +2P21

+ r(o,

0,2)jJ31331,

Qw13,

(6’)

407

17 l)PwzPZl

= q,

0, Qw12,

(f-0

Q,

1, Q3P31

= q,

LO)pm3,

with (5’) + (5”) = (5) and (6’) + (6”) = (6), respectively. Noting that ~12 + ~13 = 1 and ~21 = ~31 = 1, the following relations can be derived from these local balance equations: 7r(l 70 7

1)

7r(l

P3

= 7r(270 7O)c"'p 13’

7r(O70 7a>= n(2 70 70) ( 7r(O, 1,1) = 7r(2,0,0)

fip13 P3

71 70) = 7r(2 70 ?0)&p P2 12’

> 2 7 n-(0 72 O)= 42 7

7

0 70) ( fip12 > 2 7

Pu:!

P2 -p12P13. p2p3

Imposing the normalization condition, that the sum of probabilities of all states in the network is 1, for 7r(2,0,0) we get the following expression:

After inserting the values, we get the following results:

7r(2,0,0) = 0.103, 7r(O,O,2) = 0.148, r(l, (41) = 0.123, ~(0,2,0)

= 0.263,

~(1, I$)

= 0.165,

~(0, I, I) = 0.198.

PRODUCT-FORM QUEUEING NETWORKS

281

As can be seen in this example, the structure of the local balance equations is much simpler than the global balance equations. However, not every network has a solution of the local balance equations but there always exists a solution of the global balance equations. Therefore, local balance can be considered as a sufficient (but not necessary) condition for the global balance. Furthermore, if there exists a solution for the local balance equations, the model is then said to have the local balance property. Then this solution is also the unique solution of the system of global balance equations. The computational effort for the numerical solution of the local balance equations of a queueing network is still very high but can be reduced considerably with the help of a characteristic property of local-balanced queueing networks: For the determination of the state probabilities it is not necessary to solve the local balance equations for the whole network. Instead, the state probabilities of the queueing network in these cases can be determined very easily from the state probabilities of individual single nodes of the network. If each node in the network has the local balance property, then the following two very important implications are true: l

The overall network also has the local balance property as proven in [CHT77].

l

There exists a product-form +%,

s2, * * * ) S,)

=

solution for the network, that is: ;

[7r(S1)* n(S2)*

. . . * 7T(S,)]

)

(7.47)

in the sense that the expression for the state probability of the network is given by the product of marginal state probabilities of each individual node. The proof of this fact can be found in [Munt73]. The normalization constant G is chosen in such a way that the sum of probabilities over all states in the network equals 1. Equation (7.47) says that in networks having the local-balance property, the nodes behave as if they were single queueing systems. This characteristic means that the nodes of the network can be examined in isolation from the rest of the network. Networks of the described type belong to the class of socalled separable network or product-form networks. Now we need to examine for which types of elementary queueing systems a solution of the local balance equation exists. If the network consists of only these types of nodes then we know, because of the preceding conclusion, that the whole network has the local-balance property and the network has a product-form solution. The local-balance equations for a single node can be presented in a simplified form as follows [SaCh81]:

~(S)%-(S) = T(S -1,)*X,.

(7.48)

282

QUEUEING NETWORKS

In this equation, Pi is the rate with which class-r jobs in state S are serviced at the node, A, is the rate at which class-r jobs arrive at the node, and (S - 1,) describes the state of the node after a single class-r job leaves it. It can be shown that for the following types of queueing systems the local balance property holds [Chan72]: Type-l : M/M/ m-FCFS. The service rates for different job classes must be equal. Examples of Type-l nodes are input/output (I/O) devices or disks. Type-Z?: M/G/l-PS. The CPU of a computer system can very often be modeled as a Type-2 node. Type-3: M/G/w nodes.

(infinite

server).

Terminals can be modeled as Type-3

Type-d: M/G/l-LCFS PR. There is no practical example for the application of Type-4 nodes in computer systems. For Type-2, Type-3, and Type-4 nodes, different job classes can have different general service time distributions, provided that these have rational Laplace transform. In practice, this requirement is not an essential limitation as any distribution can be approximated as accurately as necessary using a Cox distribution. In the next section we consider product-form solutions of separable networks in more detail. Problem 7.1 Consider the closed queueing network given in Fig. 7.12. The network consists of two nodes (N = 2) and three jobs (K = 3). The

Fig. 7.12

A simple queueing network example for Problem 7.1.

service times are exponentially distributed with mean values l/pi = 5 set and l/p2 = 2.5 sec. The service discipline at each node is FCFS, and the routing probability 4 = 0.1. (a) Determine the local balance equations. (b) From the local balance equations, derive the global balance equations.

(4

Determine the steady-state probabilities tions.

using the local balance equa-

PRODUCT-FORM QUEUEING NETWORKS

7.3.3

283

Product-Form

The term product-form was introduced by [Jack631 and [GoNe67a], who considered open and closed queueing networks with exponentially distributed interarrival and service times. The queueing discipline at all stations was assumed to be FCFS. As the most important result for the queueing theory, it is shown that for these networks the solution for the steady-state probabilities can be expressed as a product of factors describing the state of each node. This solution is called product-form solution. In [BCMP75] these results were extended to open, closed, and mixed networks with several job classes, nonexponentially distributed service times and different queueing disciplines. In this section we consider these results in more detail and give algorithms to compute performance measures of product-form queueing networks. A necessary and sufficient condition for the existence of product-form solutions is given in the previous section but repeated here in a slightly different way: Local Balance Property: Steady-state probabilities can be obtained by solving steady-state (global) balance equations. These equations balance the rate at which the CTMC leaves that state with the rate at which the CTMC enters it. The problem is that the number of equations increases exponentially in the number of states. Therefore, a new set of balance equations, the so-called local balance equations, is defined. With these, the rate at which jobs enter a single node of the network is equated to the rate at which they leave it. Thus local balance is concerned with a local situation and reduces the computational effort.

Moreover, there exist two other characteristics that apply to a queueing network with product-form solution: M + M-Property (Markov Implies Markov): A service station has the M + M-property if and only if the station transforms a Poisson arrival process into a Poisson departure process. In [Munt73] it is shown that a queueing network has a product-form solution if all nodes of the network have the M + M-property. Station-Balance Property: A service discipline is said to have stationbalance property if the service rates at which the jobs in a position of the queue are served are proportional to the probability that a job enters this position. In other words, the queue of a node is partitioned into positions and the rate at which a job enters this position is equal to the rate with which the job leaves this position. In [CHT77] it is shown that networks that have the station-balance property have a productform solution. The opposite does not hold.

284

QUEUEING NETWORKS

Fig. 7.13 Relation between SB, LB, PF, and M + M.

The relation between station balance (SB), local balance (LB), productform property (PF) , and Markov implies Markov property (n/r + A4) is shown in Fig. 7.13. 7.3.4

Jackson

Networks

The breakthrough in the analysis of queueing networks was achieved by the works of Jackson [Jack57, Jack63]. He examined open queueing networks and found product-form solutions. The networks examined fulfill the following assumptions: There is only one job class in the network. The overall number of jobs in the network is unlimited. Each of the N nodes in the network can have Poisson arrivals from outside. A job can leave the network from any node. All service times are exponentially

distributed.

The service discipline at all nodes is FCFS. The ith node consists of mi 2 1 identical service stations with the service rates pi, i = 1,. . . , N. The arrival rates Xoi, as well as the service rates, can depend on the number Ici of jobs at the node. In this case we have load-dependent service rates and load-dependent arrival rates. Note: A service station with more than one server and a constant service rate pi is equivalent to a service station with exactly one server and loaddependent service rates:

(7.49)

PRODUCT-FORM QUEUEING NETWORKS

Jackson’s Theorem: for all nodes i = Eq. (7.1)), th en expressed as the nodes, that is: ;?-(h,

285

If in an open network ergodicity (A; < pi. mi) holds 1,. . . , N (the arrival rates Xi can be computed using the steady-state probability of the network can be product of the state probabilities of the individual

k2, * * * , kN)

= 7b(h)‘~2(Jc2)’

(7.50)

* * * ‘TN(kN).

The nodes of the network can be considered as independent M/M/m queues with arrival rate Xi and service rate pi. To prove this theorem, [Jack631 has shown that Eq. (7.50) fulfills the global balance equations. Thus, the marginal probabilities ri(lci) can be computed with the well-known formulae for M/M/m systems (see Eqs. (6.26), (6.27)):

(7.51)

where 7ri(O) is given by the condition

E ri(ki)

= 1:

k,=O

-’ 7 pi= Proof: t ions:

= 5

-

Ai

< 1.

(7.52)

m&i

We verify that Eq. (7.50) fulfills the following global balance equa-

Xoiy(ki)n(kl,

. . . , ki - 1,. . . , kN)

i=l

(7.53) r(k1,. . . , ki + 1,. . . , kN) N

N

+ c c q ( kj + l)/.qpgr(kl, i=l j=l

. . . , kj + 1,. . . , ki - 1, . . . , kN).

The indicator function y(ki) is given by: y(h) =

0,

ki = 0,

1, ki > 0.

(7.54)

286

QUEUEING NETWORKS

The function: (7.55) gives the load-dependent service rate multiplier. For the proof we use the following relations:

(7.56)

Aj/LiCXi(lc;) T(kl, 66s7kj + 1, *. -7 Ici - 1,. ~67ICN) _ Xi/.hjCXj(kj + 1)’ ~(JGl,...,Icj,...,~i,...,~N) If we divide Eq. (7.53) by ~(lcr,. . . , /CN) and insert Eq. (7.56), then, by using the relation y(/~i) . CX~(/C~) E oi (Ici) 7we get:

flhoi i=l

+ eN(k)pi i=l

=e (

I-

gpij j=l

N

XOif%ai

i=l

+x

x

i=l

Ai )

(h)

i

(7.57) N N Pi44 + CC TPjiXji=l

j=l

2

The first term can be rewritten as:

and the last one as: 2

2

i=l j=l

PiaiFk)

2

pjixj

= f)iCri(ki) i=l

_ 2 i=l

AOipTi(ki). a

(7.58)

Substituting these results on the right side of Eq. (7.57), we get the same result as on the left side. q.e.d. The algorithm based on Jackson’s theorem for computing the steady-state probabilities can now be described in the following three steps:

For all nodes, i = 1, . . . , IV, compute the arrival rates Xi of the open network by solving the traffic equations, Eq. (7.1).

287

PRODUCT-FORM QUEUEING NETWORKS

Consider each node i as an M/M/m queueing system. Check the ergodicity, Eq. (6.5), and compute the state probabilities and performance measures of each node using the formulae given in Section 6.5. Using Eq. (7.50), compute the steady-state probabilities overall network.

of the

We note that for a Jackson network with feedback input, processes to the nodes will, in general, not be Poisson and yet the nodes behave like independent M/M/m nodes. Herein lies the importance of Jackson’s theorem [BeMe78]. We illustrate the procedure with the following example. Example 7.4 Consider the queueing network given in Fig. 7.14, which consists of N = 4 single server FCFS nodes. The service times of the jobs at

Fig. 7.14 Open queueing network model of a computer system.

each node are exponentially distributed with respective means: -

1

= o.odsec,

tQ1

-

1

= 0.03seq

t92

The interarrival

-

1

= O.O6sec, -

1

= 0.05sec.

P4

IQ3

time is also exponentially distributed

with the parameter:

X = X04 = 4 jobs/set. Furthermore, the routing probabilities are given as follows: P12 = ~13 = 0.5,

~41 = p21 =

1,

p31 = 0.6,

p30 = 0.4.

Assume that we wish to compute the steady-state probability of state (ICI, k2, k3, k4) = (3,2,4,1) with the help of the Jackson’s method. For this we follow the three steps given previously: Compute the arrival rates from the traffic equations, Eq. (7.1): h

= X2P21

x3 = hP13

+ x3p31+ = 10,

X4P41

= 20,

x2 = hp12 x4 =

= 10,

x()4 = 4.

288

QUEUEING NETWORKS

Compute the state probabilities and important performance measures for each node. For the utilization of a single server we use Eq. (6.3): Xl p1=--0.8, I-L1

x2 p2=-=0.3, P2

p3=x3&6, CL3

p&L~. I-L4

Thus, ergodicity (pi < 1) is fulfilled for all nodes. The mean number of jobs at the nodes is given by Eq. (6.13):

j+P1=

4,

K2 = 0.429,

K3=l.5,

17*=O.a5.

1 -P1

Mean response times, from Eq. (6.15):

gil -

llm 1 -

= 0.2,

T2 = 0.043,

T3 = 0.15,

T4 = 0.0625.

Pl

The mean overall response time of a job is given by using Little’s theorem in the following way: T=

K x = $Fi

= 1.545.

1=1

Mean waiting times, from Eq. (6.16): w1 = fd$

= 0.16,

w2 = 0.013,

w3 = 0.09,

w4 = 0.0125.

Mean queue lengths, from Eq. (6.17):

~~ = - Pi 1 -

= 3.2, Q2 = 0.129,

Q3

=0.9,

Q4

=0.05.

Pl

The necessary marginal probabilities

can be computed using Eq. (6.12):

nr(3) = (1 - pl)p; = 0.1024,

7r2(2) = (1 - p2)p; = 0.063,

~~(4) = (1 - p3)p; = 0.0518,

7r4(1) = (1 - p4)p4 = 0.16.

Computation

of the state probability

7r(3,2,4,1)

using Eq. (7.50):

7r(3,2,4,1) = 7rr(3)* 7r2(2)* 7r3(4)* n-4(1) = 0.0000534. 7.3.5

Gordon/Newell

Networks

Gordon and Newell [GoNe67a] considered closed queueing networks for which they made the same assumptions as in open queueing networks, except that

PRODUCT-FORM QUEUEING NETWORKS

289

no job can enter or leave the system (X0; = X;e = 0). This restriction means that the number K of jobs in the system is always constant:

Thus, the number of possible states is finite, and it is given by the binomial coefficient:

which describes the number of ways of distributing K jobs on N nodes. The theorem of Gordon/Newell says that the probability for each network state in equilibrium is given by the following product-form expression: (7.59) Here G(K) is the so-called normalixation constant. It is given by the condition that the sum of all network state probabilities equals 1:

G(K) =

C fiF,(ki)5 ki=K '=' a=1

The Fi(ki) are functions that correspond to the state probabilities the ith node and are given by:

(7.60)

ri(lci) of

(7.61) where the visit ratios ei are computed using Eq. (7.5). The function pi(Ici) is given by:

pi(ki) =

ki!, mi! * rnfamrni,

1,

ki I mi, Ici > mi, mi = 1.

(7.62)

For various applications a more general form of the function Fi (Ici) is advantageous. In this generalized function, the service rates depend on the number of jobs at the node. For this function we have:

ki e; Fi(ki) = Ai(Q’

(7.63)

290

QUEUEING NETWORKS

with

(7.64)

With relation (7.49) it can easily be seen that the case of constant service rates, Eq. (7.61), is a special case of Eq. (7.63). Proof: Gordon/Newell have shown in [GoNe67a] that Eq. (7.59) fulfills the global balance equations (see Eq. (7.53)) that have now the following form:

(7.65)

where the left side describes the departure rate out of state (ki, . . . , kN> and the right side describes the arrival rate from successor states into this state. The function y(ki) and cxi(ki) are given by Eqs. (7.54) and (7.55). We define now a variable transformation as follows: @l,.

. . , kN) =

Q(h,--.,k~) fi Pi(k)

i=l

(7.66) ’

and: +l,

. . . , ki - 1,. . . , kj + 1,. . . , kN) =

-$&Q(k, 3

,...,

ki-l,...,Ic,+l,...$N) . fi

Pi(k)

i=l

If we substitute these equations into Eq. (7.65) we get:

(7.67)

291

PRODUCT-FORM QUEUEiNG NETWORKS

This equation can be simplified using the relation r(lci)cri(ki)

E ai(lci):

N

c a@i)piQ(h, -a.7

kN)

=

i=l N

(7.68)

N

cc ai(k)pjpjiQ(h,. ..,h- 1,...,kj+1,...,h), j=l

i=l

and Q(rCr,. . . , kN) can be written in the form: (7.69) with the relative utilization xi = ei/pi (Eq. (7.6)). Here c is a constant. Substituting this into (7.68) we have:

i=l

= 5

2

cui(~i)pjpji(x~l

. . . xy

. . . x;j+l

. . . x$cJ) . c

This expression can be rewritten as:

gm (pi-gpjpji$ =0. Since at least one ai (ki) will be non-zero, it follows that the factor in the square brackets must be zero. Thus, we are led to consider the system of linear algebraic equations for xi: N /LiZi

=

pjxjPji,

c j=l

where xi = e;/pi (Eq. (7.6)) and: N ei =

c j=l

ejPji*

292

QUEUEING NETWORKS

This is the traffic equation for closed networks (Eq. (7.5)). That means that Eq. (7.69) is correct and with Eqs. (7.66) and (7.61) we obtain Eq. (7.59).q.e.d. Thus the Gordon/Newell theorem yields a product-form solution. In the general form it says that the state probabilities r(lcl, kz, - - . , kN) are given as the product of the functions Fi (ki), i = 1. . . , N, defined for single nodes. It is interesting to note that if we substitute in Eq. (7.59) Fi(lci) by Li. Fi(ki), 1 < i 5 N, then this has no influence on the solution of the state probabilities , kN> as long as Li is a positive real number. Furthermore, the +h,... use of X.ei, i = l,... , N, with an arbitrary constant X > 0, has no influence on the results because the visit ratios ei are relative values [ShBu77]. (Also see Problems 2 and 3 on page 444 of [Riv82].) The Gordon/Newell method for computing the state probabilities can be summarized in the following four steps: Compute the visit ratios ei for all nodes i = 1,. . . , N of the closed network using Eq. (7.5). For alli = l,... , N, compute the functions Fi(ki) using Eq. (7.61) or Eq. (7.63) (in th e case of load-dependent service rates). Compute the normalization

constant G(K) using Eq. (7.60).

Compute the state probabilities of the network using Eq. (7.59). From the marginal probabilities, which can be determined from the state probabilities using Eq. (7.16), all other required performance measures can be determined. An example of the application of the Gordon/Newell

theorem follows:

Example 7.5 Consider the closed queueing network shown in Fig. (7.15) with N = 3 nodes and K = 3 jobs. The queueing discipline at all nodes is FCFS. The routing probabilities are given as follows:

P12 = 0.3,

p21 = 0.2, P22 = 0.3,

P13 = 0.1,

P23 =

~11 = 0.6,

0.5,

The service time at each node is exponentially ~1 =

0.8 se?,

This network consists of

p2 = 0.6 set-l,

p32

0.4, = 0.1,

p33

= 0.5.

P31 =

distributed with the rates: p3 = 0.4 set-l .

(7.70)

PRODUCT-FORM QUEUEING NETWORKS

Fig. 7.15

293

Closed queueing network.

states, namely: (3,0, O),

(2,L o),

(270, I>,

(k2, O),

(Lb I>>

(l,O, 2),

(073, O),

(0,2,1>,

(0,1,2),

(O,O?3).

We wish to compute the state probabilities using the Gordon/Newell orem. For this we proceed in the following four steps: r-1

-1

Determine the visit ratios at each node using Eq. (7.5): el = elpll

+

e2p21

+

e3p31

=

I,

e2

=

elpl2

+

e2p22

+

e31332

=

0.533,

e3

=

elpl3

+e2p23

=

0.733.

+e3p33

Determine the functions Fi(ki) for i = 1,2,3 using Eq. (7.61):

WO = (el/pl)o = 1,

W) = (e1/d = 1.25,

Fr(2) = (el/p1)2 = 1.5625,

Fr(3) = (el/p1)3 = 1.953,

and correspondingly: F2(0) = 1, F?(O) = 1, m]

Fz(l) = 0.889, F3(1) = 1.833,

F2(2) = 0.790, Fs(2) = 3.361,

Determine the normalization G(3) =

Fr(3)F2(0)&(0) + ~~(~)~2(2)~3(0)

&(3) = 0.702, Fs(3) = 6.162.

constant using Eq. (7.60):

+ Fr(2)&(1)Fs(O) + h(l)F&)F&)

+ Fr(2)FZ?(O)F$) + Fl(1)h(o)F3(2)

the-

294

QUEUEING NETWORKS

+ pl(“)F2(3)~3(o)

+ Fl(o)F2(2)F3(1)

+ Fl(O)&(O)F3(3)

= 24.733.

Determine the state probabilities rem, Eq. (7.59):

+

using the Gordon/Newell

1

= -q3)*qo)* G(3)

n(3,0,0)

F3(0)

0) = L~r(2).~2(l)+3(0) G(3)

G,L

F1(0)&(1)F3(2)

=

theo-

0.079,

= 0.056.

In the same way we compute: 7r(2,0,1)

= 0.116,

~(0,3,0)

= 0.028,

n(1,2,0) x(0,2,1)

= 0.040, = 0.058,

?r(l, 1,l) = 0.082, ~(0, 1,2) = 0.121,

T&0,2) = 0.170, r(O,O, 3) = 0.249.

Using Eq. (7.16), all marginal probabilities can now be determined: q(o) = 7r(o, 3,0) + n(O,2,1) + 7r(O,1,2) + 7r(O,0,3) = 0.457, 7rr(l) = 7r(l, 2,O) + 7r(l, 1,l) + T(l,O, 2) = 0.292, ~(2)

= n(2,1,0)

+ n(2,0,1)

= 0.172,

7r1(3) = 7r(3,0,0)

= 0.079,

~~(0) = 7r(2,0,1)

+ r(l,O, 2) + r(O,O, 3) + n(3,0,0)

= 0.614,

T2(1) = n(2,1,0) + 7r(l, 1,l) + T(O,l, 2) = 0.259, 7r2(2) = x(1,2,0)

+ 7r(O,2,1) = 0.098,

T2(3) = x(0,3,0)

= 0.028,

~~(0) = 7r(3,0,0)

+ n(2,1,0)

+ r&2,0)

+ n(O,3,0) = 0.203,

7r3(1) = 7r(2,0,1) + ~(1, 1,l) + ~(0,2,1) = 0.257, ~~(2) = r(l,O, 2) + ~(0,1,2) = 0.291, 7r3(3) = 7r(O,

0,3) = 0.249.

For the other performance measures, we get: l

Utilization

at each node, Eq. (7.19): p1 = 1 - 7rr(O) = 0.543,

l

p2 = 0.386,

p3 = 0.797.

Mean number of jobs at each node, Eq. (7.26):

K1= &q(k) k=l

= 0.873 &, = 0.541, K3= 1.585.

295

PRODUCT-FORM QUEUEING NETWORKS

Throughputs at each node, Eq. (7.23):

l

Xl =mlplpl

= 0.435, X2 = 0.232, X3 = 0.319.

Mean response time at each node, Eq. (7.43):

l

T, = -Kl = 2.009, T2 = 2.337, T3 = 4.976. h

7.3.6

BCMP

Networks

The results of Jackson and Gordon/Newell were extended by Baskett , Chandy, Muntz, and Palacios in their classic article [BCMP75], to queueing networks with several job classes, different queueing strategies, and generally distributed service times. The considered networks can be open, closed, or mixed. Following [BrBa80], an allowed state in a queueing model without switching is characterized by four conditions:

class

1. The number of jobs in each class at each node is always nonnegative, i.e.:

2. For all jobs the following condition must hold: iii, > 0

if there exists a way for class-r jobs to node i with a non-zero probability.

3. For a closed network, the number of jobs in the network is

by:

4. The sum of class-r jobs in the network is constant at any time, i.e.:

K, = gki, given

= const.

l
i=l

If class switching is allowed, then Conditions 1-3 are fulfilled, but Condition 4 can not be satisfied because the number of jobs within a class is no longer constant but depends on the time when the system is looked at and can have the values k E {O, . . . , K}. In order to avoid this situation, the concept of chains is introduced.

296

QUEUEING NETWORKS

7.3.6.1 The Concept of Chains Consider the routing matrix P = bir,jS], i,j = l,..., N, and T, s = 1,. . . , R of a closed queueing network. The routing matrix P defines a finite-state DTMC whose states are pairs of form (i, r) (node number i, class index r). We call (i,r) a node-class pair. This state space can be partitioned into disjoint sets Ii, i = 1,. . . , U:

r = rl + r2 + . . . + ru, where Ii is a closed communicating

class of recurrent states of the DTMC.

Pl

M-l

fig. 7.16

p2

0

Modified routing matrix.

With a possible relabeling of nodes, we get the routing matrix of Fig. 7.16. Where submatrices Pi contains the transition probabilities in set I’i and each Pi is disjoint and assumed to be ergodic. Let chain Ci denote the set of job classes in ri. Because of the disjointness of the chains Ci, it is impossible for a job to switch from one chain to another. If a job starts in a chain, it will never leave this chain. With the help of this partitioning technique, the number of jobs in each chain is always constant in closed networks. Therefore, Condition 4 for queueing networks is fulfilled for the chains. Muntz [Munt73] proved that a closed queueing network with U chains is equivalent to a closed queueing network with U job classes. If there is no class switching allowed, then the number of chains equals the number of classes. But if class switching is allowed, the number of chains is smaller than the number of classes. This means that the dimension of the population vector is reduced. An extension of the chains concept to open networks is possible. The procedure to find the chains from a given solution matrix is the following:

PRODUCT-FORM QUEUEING NETWORKS

297

Construct R sets that satisfy the following condition:

node-class pair (j, s) can be reached from pair (i, r) in a finite number of steps

7

Eliminate all subsets and identical sets so that we have: U 5 R

sets to obtain the chains

Cl,. . . , Cu.

Compute the number of jobs in each chain

The set of all possible states in a closed multiclass network is given by the following binomial coefficient:

N+$[+K,*-I N. IC,l -

’

I

where IC, ( is the number of elements in C,, llq
r E Cq, 2%

= PO,ir +

C

ejsPjs,ir

for

i = l,...N,

(7.71)

lLqlU,

SEC,

j=l,...,N

and for closed networks:

-Sir =c

SEC, j=l,...,N

r E c,, ejsPjs,ir

for

i= l,...N, lSq
(7.72)

298

QUEUEING NETWORKS

Example

7.6

Let the following routing matrix P be given: (1.1)

=

(292)

(293)

(3.1)

(3,2)

(393)

0

0

0.3

0

0

0.3

0

0.3

0

0

0

0

0

0

0.7

0

(133)

0

0

0

0

0

0.3

0

0

0.7

(2,1)

0

0

0

0

1

0

0

0

0

(292)

0.3

0

0

0

0

0

0.7

0

0

(2>3)

0

0

0.3

0

0

0

0

0

0.7

(331)

1.0

0

0

0

0

0

0

0

0

(3.2)

0

0

0

1.0

0

0

0

0

0

(3,3)

, 0

0

0.4

0

0

0.6

0

0

0

(1?2)

p

(221)

0.4

(1.1)

’ 0

(1,2) (1>3)

By using the chainin!

;echnique we get:

EI =

{1,2},

E2

=

{Q},

E3

= {3),

meaning that R = 3 classes are reduced to U = 2 chains. By eliminating the subsets and identical sets we get two chains: Cl = {1,2},

c2 = (3).

Then the reorganized routing matrix P’ has the form: (19 1) (13 1)

p’ =

’ 0

(1?2)

(2,1)

(2.2)

(3,l)

(3,2)

(1.3)

(2.3)

(393)

0.4

0

0.3

0

0.3

0

0

0

(13 2)

0.3

0

0

0

0

0.7

0

0

0

(2>1)

0

0

0

1.0

0

0

0

0

0

(272)

0.3

0

0

0

0.7

0

0

0

0

(3,1)

1.0

0

0

0

0

0

0

0

0

(312)

0

0

1.0

0

0

0

0

0

0

(193)

0

0

0

0

0

0

0

(223)

0

0

0

0

0

0

0.3

0

0

0

0

0

0.4 0.6

(3.3)

,o

0.3 0.7 0

0.7 0

If a job starts in one of the chains Cl or C2, then it can not leave them. Because of the switch from classes to chains, it is necessary to transfer class measures into chain measures [BrBa80] (ch ain measures are marked with a *):

PRODUCT-FORM QUEUEING NETWORKS l

299

Number of visits: In a chain a job can reach different node-class pairs (i, j) (i: node index, j: class index) where: C eir r-4, * eiq = Gel,’

(7.73)

rd2,

l

The number of jobs per class is constant in a chain, but within the chain jobs can change their class. Therefore, if a job of chain q visits node i, it is impossible to know to which class it belongs because we consider a chain to be a single entity. If we make the transformation class

chain,

+

the information about the class of a job is lost. Because different job classes have different service times, this missing information is exchanged by the scale factor cv. For the service rate in a chain we get: 1 siq = * = c I-L. vl r&, ai,

=

-

gireis

Sir

(7.74)

’ air 9

(7.75)

’

SEC,

In Chapter 8 we introduce several algorithms to calculate the performance measures of single and multiclass queueing networks without class switching (such as mean value analysis or convolution). Using the concept of chains, it is possible to also use these algorithms for queueing networks with class switching. To do so, we proceed in the following steps: Calculate the number of visits ei, in the original network. Determine the chains Ci . . . Cu, and calculate the number of jobs Kg* in each chain. Compute the number of visits ezq for each chain, Eq. (7.73). Determine the scale factors air, Eq. (7.75). Calculate the service times szq for each chain, Eq. (7.74). Derive the performance measures per chain [BrBa80] with one of the algorithms introduced later (mean value analysis, convolution, etc.). Calculate the performance measures per class from the performance measures per chain: Tir(K*)

= sir * (1 + Ki(K*

- 1,)) ,

T E Cq

300

QUEUEING NETWORKS

Ai,

= aiT . A$,

pi&*)

= sir - kr(K*),

where: Population vector containing the number of jobs in each chain, K* - 1, : K” with one job less in chain q, TjT,(K* - lq) : Mean number of jobs at node i if the number of jobs in the chains is given by (K* - 14).

K* = (K;,...,K;):

All other performance measures can easily be obtained using the formulae of Section 7.2. In the following section we introduce the BCMP theorem, which is the basis of all analysis techniques to come. If class switching is not allowed in the network, then the BCMP theorem can be applied directly. In the case of class switching, it needs to be applied in combination with the concept of chains. 7.3.6.2 BClVlP Theorem The theorems of Jackson and Gordon/Newell have been extended by [BCMP75] to networks with several job classes and different service strategies and interarrival/service time distributions, and also to mixed networks that contain open and closed classes. The networks considered by BCMP must fulfill the following assumptions: l

Queueing discipline: The following disciplines are allowed at network nodes: FCFS, PS, LCFS-PR, IS (infinite server).

l

Distribution of the service times: The service times of an FCFS node must be exponentially distributed and class-independent (i.e. pi1 = pi2 = . . . = ,&R = pi), while PS, LCFS-PR and IS nodes can have any kind of service time distribution with a rational Laplace transform. For the latter three queueing disciplines, the mean service time for different job classes can be different.

l

Load-dependent service rates: The service rate of an FCFS node is only allowed to depend on the number of jobs at this node, whereas in a PS, LCFS-PR and IS node the service rate for a particular job class can also depend on the number of jobs of that class at the node but not on the number of jobs in another class.

l

Arrival processes: In open networks two kinds of arrival processes can be distinguished from each other. Case 1: The arrival process is Poisson where all jobs arrive at the network from one source with an overall arrival rate X, where X can depend

PRODUCT-FORM QUEUEING NETWORKS

301

on the number of jobs in the network. The arriving jobs are distributed over the nodes in the network in accordance to the probability PO,+. where:

E i=l

PO,ir = 1.

r=l

Case 2: The arrival process consists of U independent Poisson arrival streams where the U job sources are assigned to the U chains. The arrival rate X, from the uth source can be load dependent. A job arrives at the ith node with probability PO,+.so that:

c

PO,ir = 17

for all u= l,...,U.

rEC, i=l

I..., N

These assumptions lead to the four product-form node types and the local balance conditions for BCMP networks (see Section 7.3.2), that is: Type-l:

-/M/m

Type-3:

-/G/CO (IS)

- FCFS

Type-2: Type-4:

-/G/l -/G/l

- PS - LCFS PR

Note that we use -/M/ m notation since we know that, in general, the arrival process to a node in a BCMP network will not be Poisson. The BCMP Theorem says that networks with the characteristics just described have product-form solution: For open, closed, and mixed queueing networks whose nodes consist only of the four described node types, the steady-state probabilities have the following product-form: (7.76)

where, G(K) = the normalization constant, d(S) = a function of the number of jobs in the network, S = (Si, . . . , S,) and: K(S)-1 w

open network with arrival process 1,

i=o u K,(S)--1 n J-J WL

open network with arrival process 2,

r-J

d(S) =

I

u=l

1,

i=o

closed networks.

QUEUEING NETWORKS

302

fi(Si)

= a f uric t ion which depends on the type and state of each nodei

Type-l, Type-% (7.77)

Type-3, Type-4. Variables in for Eq. (7.77) have the following meanings: szj : Pir1: uir :

Class of the job that is at the jth position in the FCFS queue. Mean service rate in the Zth phase (I = 1, . . . , uir) in a Cox distribution (see Chapter 1). Maximum number of exponential phases. l-l

n airj, probability that a class-r job at the ith node reaches the Zth j=o service phase (A,,1 = 1 because of aiT0 = 1). CLirjZ Probability that a class-r job at the ith node moves to the (j + 1)th phase. b-1 1 Number of class-r jobs in the Zth phase of node i. Ail-1

:

For the load-dependent case, fi (Si) is of the form: fi(Si)

=

-A- Ici iifZiszj. ( Pi > j=l

Proof: The proof of this theorem is very complex and therefore only the basic idea is given here (for the complete proof see [Munt72]). In order to find a solution for the steady-state probabilities r(S), the following global balance equations have to be solved: state transition rate ] n(s) 1from state S

with the normalization

98

state transition rate from state S to state S

17 (7.78)

condition:

CT(S) = 1. S

(7.79)

PRODUCT-FORM QUEUEING NETWORKS

303

Now we insert Eq. (7.76) into Eq. (7.78) to verify that Eq. (7.78) can be written as a system of balance equations that [Chan72] calls local balance equations. All local balance equations can be transformed into a system of N - 1 independent equations. To get unambiguity for the solution the normalization condition, Eq. (7.79), has to be used. Now we wish to give two simplified versions of the BCMP theorem for open and closed networks. BCMP Version 1: For a closed queueing network fulfilling the assumptions of the BCMP theorem, the steady-state state probabilities have the form:

n(S1,. ..)S,)= & fi fiw

(7.80)

2=1

where the normalization

constant is defined as: (7.81)

and the function Fi(Si) is given by:

ki

1 . nR -efJT, Ici,! r=l

ZqSi)

=

Type-l,

Type-W,

(7.82)

Type-3.

The quantity Ici = 5 Icir gives the overall number of jobs of all classes at r=l

node i. The visit ratios ei, can be determined with Eq. (7.72), while the function ,&(lci) is given in Eq. (7.62). For FCFS-nodes (mi = 1) with load-dependent service rates pi(j) we get: (7.83)

BCMP Version 2: For an open queueing network fulfilling the assumptions of the BCMP theorem and load-independent arrival and service rates, we have:

304

QUEUEING NETWORKS

where:

(1-Pi)P;“,

Type-1,2,4 (mi = l),

7ri(ki) =

(7.85) e---pz& lci! ’

Type-3,

with: &-F,

i

Type-l

(mi = l), (7.86)

pir = A, E,

Type-2,3,4.

Furthermore we have:

ri, = pir

1 - pi*

(7.87)

For Type-l nodes with more than one service unit (mi > 1)) Eq. (6.26) can be used to compute the probabilities pi. For the existence of the steady-state probabilities ri(~?i), ergodicity condition (pi < 1) has to be fulfilled for all i, i = l,...,N. The algorithm to determine the performance measures using the BCMP theorem can now be given in the following five steps: -1 Compute the visit ratios ei, for all i = 1,. . . , N and r = 1,. . . , R using Eq. (7.71). m/

Compute the utilization

of each node using Eq. (7.86)

-1 Compute the other performance measures with the equations given in Section 7.2. w

Compute th e marginal probabilities of the network using Eq. (7.85).

-1

Compute th e st at e probabilities

using Eq. (7.84)

Example 7.7 Consider the open network given in Fig. 7.17 with N = 3 nodes and R = 2 job classes. The first node is of Type-2 and the second and

PRODUCT-FORM QUEUEING NETWORKS

third nodes are of Type-4. with the rates: ~11 =

The service times are exponentially

8sec-‘,

The interarrival

~22 =

distributed

~31 = 16 see-‘,

~21 = 12 set-‘,

~12 = 24sec-',

305

32secA1,

~32

= 36sec-‘.

times are also exponentially distributed with the rates: X1 = X2 = 1 job/set.

q---q----

CL3 cl Sink

Fig. 7.17 Open queueing network.

The routing probabilities are given by: PO,11 = 1, P11,21

=

P21,11

0.4, p 21,31

P31,11 = 0.5,

P11,31 = 0.3, Pll,O

=

= 0.6, = 0.4,

0.3,

p31,21 =

0.5,

PO,12 =

1,

P12,22 = 0.3, Pl2,32 =

0.6,

p12,O = 0.1,

P22,12

= 0.7,

p22,32 = 0.3, P32,12

= 0.4,

1)32,22 =

0.6,

which means that class switching is not allowed in the network. We wish to compute the probability for the state (Ici, Icz,Ics) = (3,2,1) by using the BCMP theorem, Eq. (7.84). ~~~~ Compute the visit ratios ei, for all i = 1,. . . , N and r = 1,. . . , R using Eq. (7.71). ell

= POJI

+ elml,ll

+ e2lp21,ll

+ e3lp31,ll

= 3.333,

e21 = P0,21 + fmm,21

+ e211321,21 + e31p31,21

= 2.292,

e31 = P0,31 + elml,31

+ e2lp21,31

= 1.917.

+ e311331,31

In the same way we get: el2 = 10,

e22 =

8.049,

es2 = 8.415.

306

QUEUEING NETWORKS

Compute the utilization p1 = J@L

+ he12

= ~11 + p12 = 0.833,

CL12

I-L11 P2 =

of each node using Eq. (7.86):

X1 e21 + X2% CL21 P22

= p21 + p22 = 0.442,

P3=xle31+x2~=pJl+p32=o.354. P31

I-L32

Compute the other performance measures of the network. In our case we use Eq. (7.87) to determine the mean number of jobs at each node:

El1 = *1 El2

= 2.5,

Pl

= 2.5, 7721 = *1 K22

= 03,

K32

P2

p3

= 0.362.

Determine the marginal probabilities

~~~

n1(3) = (I-

pl)p;

7r3(1) = (l-

p3)p3 = 0.2287.

= 0.0965,

using Eq. (7.85):

7r2(2) = (I-

Compute th e st,at,e probabilities 7r(3,2,1)

= 0.186, 7731 = -1 p31

= 0.342,

p2)p5 = 0.1093,

for the network using Eq. (7.84):

= 7r1(3). 7r2(2). n3(l) = 0.00241.

An example of the BCMP theorem for closed networks, Eq. (7.80), is not given in this chapter. As shown in Example 7.7, the direct use of the BCMP theorem will require that all states of the network have to be considered in order to compute the normalization constant. This is a very complex procedure and only suitable for small networks because for bigger networks the number of possible states in the network becomes prohibitively large. In Chapter 8 we provide efficient algorithms to analyze closed product-form queueing networks. In these algorithms the states of the queueing network are not explicitly involved in the computation and therefore these algorithms provide much shorter computation time. Several researchers have extended the class of product-form networks to nodes of the following types: l

l

In [Spir79] the SIR0 ( service in random order) is examined and it is shown that -/M/l-SIR0 nodes fulfill the local balance property and therefore have a product-form solution. In [Noet79] the LBPS (last batch processor sharing) strategy is introduced and it is show that -/M/l-LBPS node types have product-form solutions. In this strategy the processor is assigned between the two last

PRODUCT-FORM QUEUEING NETWORKS

307

batch jobs. If the last batch consists only of one job, then we get the LCFS-PR strategy and if the batch consists of all jobs, we get PS. l

In [ChMa83] the WEIRDP-strategy (weird, parameterized strategy) is considered, where the first job in the queue is assigned 100 . p % of the processor and the rest of the jobs are assigned 100 . (1 - p) % of the processor. It is shown that -/M/l-WEIRDP nodes have product-form solution.

l

In [CHT77] it is shown that -/G/l-PS, -/G/co-IS, and -/G/l-LCFSPR nodes with arbitrary differentiable service time distribution also have product-form solution.

Furthermore, in [Tows80] and [Krze87], the class of product-form networks is extended to networks where the probability to enter a particular node depends on the number of jobs at that node. In this case, we have so-called load-dependent routing probabilities. Problem 7.2 Consider an open queueing network with N = 3 nodes, FCFS queueing discipline, and exponentially distributed service times with the mean values: -

1

= 0.08 set,

-

CL1

1

= 0.06 set,

l-42

-

1

= 0.04sec.

CL3

Only at the first node do jobs arrive from outside with exponentially distributed interarrival times and the rate Xar = 4 jobs/set. Node 1 is a multiple server node with ml = 2 server, nodes 2 and 3 are single server nodes. The routing probabilities are given as follows: Pll

= 0.2,

p21

p12 = 0.4, p13 =

=

p30 =

1,

p31 =

0.5,

0.5,

0.4.

(a) Draw the queueing network. (b) Determine the steady-state probability

for the state k = (4,3,2).

(c) Determine all performance measures. Note: Use the Jackson’s theorem. Problem 7.3 Determine the CPU utilization and other performance measures of a central server model with N = 3 nodes and K = 4 jobs. Each node has only one server and the queueing discipline at each node is FCFS. The exponentially distributed service times have the respective means: -

1

Pl

= 2 msec,

-

1

P2

= 5 msec,

-

1

I_L3

= 5 msec,

308

QUEUEING NETWORKS

and the routing probabilities are: Pll

=

0.3,

0.5,

p12 =

p21 = p31 = 1.

0.2,

p13 =

Consider a closed queueing network with K = 3 nodes, N = Problem 7.4 3 jobs, and the service discipline FCFS. The first node has ml = 2 servers and the other two nodes have one server each. The routing probabilities are given by: PII = 0.6,

1321

= 0.5,

p31 =

P12 = O-3,

p22

=

0.0, = 0.5,

p32

p23

p12 = o-1,

The service times are exponentially

0.4,

= 0.6,

p33 = 0.0.

distributed with the rates

pl = 0.4sec-l 7 /A:! = 0.6sec-1 7

0.3sec-l.

/L3 =

(a) Draw the queueing network. (b) What is the steady-state probability

that there are two jobs at node 2?

(c) Determine all performance measures. (d) Solve this p ro bl em also with the local and global balance equations respectively and compare the solution complexity for the two different methods. Note: Use the Gordon/Newell

Theorem for parts (a), (b) and (c).

Problem 7.5 Consider a closed queueing network with N = 3 nodes, K = 3 jobs, and exponentially distributed service times with the rates: pl =

0.72 set-l,

,92 = 0.64sec-’ 7

j.~3

= lsec-l,

and the routing probabilities: 1331= 0.4,

p32

= 0.6,

~13 = p23 =

1.

Determine all performance measures. Problem

7.6

Consider the following routing matrix P:

P

(Ll)

(172)

(1,3)

(174)

(271)

(2c4

(2>3)

(24)

(1,l)

0.5

0

0

0

0.25

0

0.25

0

y; (114) (2,l) (2,2) (2,3) (2,4)

0.5 0

0.5 0 0 0::

0 0 0 0 1 0

0.5 0 0.5 0 0 0 1

0 0 0.5 0 0 0

0 0.25 005

0.5 0 0 0 0 0 0

0 0.25 0 0 0 0

OY5 0 0 0

0 0

0 0

PRODUCT-FORM QlJEUElNG NETWORKS

309

At the beginning, the jobs are distributed as follows over the four different classes:

(a) Determine the disjoint chains. (b) Determine the number of states in the CTMC underlying the network. (c) Determine the visit ratios. Consider an open network with N = 2 nodes and R = 2 job Problem 7.7 classes. Node 1 is of Type-2 and node 2 of Type-4. The service rates are pll

=

4sec-‘,

~12 = 5sec-‘,

~21 = Gsec-‘,

~22 = 2sec-‘,

and the arrival rates per class are Xr=&=lsec-l. The routing probabilities are given as: PO,11 = PO,12 =

p11,21 = 0.6,

1,

P21,ll

~12,22

= P22,12

=

With the BCMP theorem, Version 2, determine (a) All visit ratios. (b) The utilizations

of node 1 and node 2.

(c) The steady-state probability

1,

= 0.4 PII,O = 0.4,

for state (2,1).

p12,o = 0.6.

8 Algorithms for Product-Form Networks Although product-form solutions can be expressed very easily as formulae, the computation of state probabilities in a closed queueing network is very time consuming if a straightforward computation of the normalization constant using Eq. (7.3.5) is carried out. As seen in Example 7.7, considerable computation is needed to analyze even a single class network with a small number of jobs, primarily because the formula makes a pass through all the states of the underlying CTMC. Therefore we need to develop efficient algorithms to reduce the computation time [Buze71]. Many efficient algorithms for calculating performance measures of closed product-form queueing networks have been developed. The most important ones are the convolution algorithm and the mean value analysis (MVA) [ReLa80]. The convolution algorithm is an efficient iterative technique for calculating the normalization constant, which is needed when all performance measures are computed using a set of simple equations. In contrast, the mean value analysis is an iterative technique where the mean values of the performance measures can be computed directly without computing the normalization constant. We also introduce the RECAL algorithm [CoGe86] (recursion by chain algorithm), which is well suited for networks with many job classes. The fourth and last method for analyzing product-form networks presented in detail is the so-called flow-equivalent server method [CHW75b, ABP85]. This method is well suited when we are especially interested in computing the performance of a single station or a part of the network. There are several other algorithms that we do not cover in detail due to space limitations. Based on the mean value analysis, [ChSa80] and [SaCh81] developed an algorithm called LBANC (local balance algorithm for normal311

312

ALGORITHMS FOR PRODUCT-FORM NETWORKS

Fig, 8.1

Flowchart describing sections of Chapters 8 and 9.

izing constants). This algorithm iteratively computes the normalization constant and the performance measures. It is very well suited for networks with a small number of nodes but a large number of jobs. The CCNC algorithm (coalesce computation of normalizing constants) [ChSa80] is especially used when storage space is limited. That the convolution algorithm, the mean value analysis, and LBANC can be derived from each other has been shown by [Lam81]. The DAC algorithm (distribution analysis by chain) [SoLa89] is used to directly compute the state probabilities of a product-form queueing network. A generalization of the MVA to higher moments is introduced in [Stre86]. Other algorithms for determining the normalization constant are the algorithms of [Koba78] and [Koba79] (w h ic h are based on the enumeration of polynomials of Polya), and the partial fraction method of [Moor72]. For very large networks, execution time of the preceding algorithms is not acceptable. Many approximation algorithms have been developed for this purpose. Exact and approximate algorithms are discussed in this chapter and in Chapter 9 and are summarized in Fig. 8.1.

THE CONVOLUTION ALGORITHM

8.1

THE

CONVOLUTION

313

ALGORITHM

The convolution algorithm was one of the first efficient algorithms for analyzing closed product-form queueing networks and is still in use today. The name of this technique reflects the method of determining the normalization constant G(K) from the functions Fi(Ici), which is similar to the convolution of two probability mass functions. Once the normalization constant is computed, the system performance measures of interest can be easily derived [Buze71]. We recall that the convolution operator @ is defined in the following way: Let A, B, C be vectors of length K + 1. Then the convolution C = A @ B is defined as:

C(k) = &A(j) j=o

. B(k - j),

k = 0, . . . , K.

(84

First we consider the convolution algorithm for single class closed queueing networks. Then this technique is extended to multiclass closed queueing networks. 8.1.1

Single

Class Closed

Networks

According to the BCMP theorem, Eq. (7.80), the state probabilities of a closed single class product-form queueing network with N nodes can be expressed in the following way:

where the functions I?i (Ici) are defined in Eq. (7.61), and:

is the normalization constant of the network. The computation of G(K) is carried out by iterating over the number of nodes in the network and over the number of possible jobs at each node. For this purpose the following auxiliary functions G, (Ic) , n = 1, . . . , N and /k = 0, . . . ) K are defined as follows:

The desired normalization

constant is then:

G(K) = GN(K).

W)

314

ALGORITHMS FOR PRODUCT-FORM NETWORKS

From Eq. (8.2) for n > 1, it follows that:

For n = 1 we have: k = l,...,

Gl(k) = Fl(k),

K.

(8.5)

The initial condition is of the form: Gn(0) = 1,

n= l,...,

N.

(8.6)

The convolution method for computing the normalization constant G(K) is fully determined by Eqs. (8.4) to (8.6). The (K + 1)-dimensional vectors:

,

Gn =

n = 1,. . . , N,

are therefore determined by the convolution:

F,=

The computation of G,(k) is easily visualized as in Fig. 8.2. In this figure we show the values of the functions needed in the calculation of GN (K). The real objective of the algorithm is to determine the last value in the last column because this value is the normalization constant G(K) that we are looking for. The values GN (Ic) for k = 1, . . . , K - 1 in the last column are also useful in determining the system performance measures. Buzen [Buze71, Buze73] has developed algorithms that are based on the normalization constant G(K) and the functions Fi(ki) to compute all important performance measures of the queueing network without needing the underlying CTMC state probabilities. Computation of the performance measures using the normalization constant is now illustrated.

THE CONVOLUTION ALGORITHM 11

... **a

n-l

0

1

Gn-1(0)(%(k)) Gn-l(O)(~Fn(k))

1

Fl(1)

... ...

I

I

k-l

Fr.(k-1) Fr.(k-1)

k

Fl(k)

K

FI (K)

... ...

Gn-l(l)(+Fn(k Gn-l(l)(*Fn(k

Gn-l(k

-

...

n

I

1

-- 1)) 1))

N 1 = G(1)

Gn(l)

l)(*Fn(l))

c

Gn-l(k)(.Fn(O))

+

Gn(k)

*.*

Gn (W

Fig. 8.2

315

Computation

Gv(k) Gv(k)

&v(K)

= = G(k) G(k)

= G(K)

of G,(k).

(a) The marginal probability that there are exactly ki = !Cjobs at the ith node is given by Eq. (7.16): n$c) =

c E

7r(/kl,...,k~).

kj=K

3=1

&k,=k

If we substitute Eq. (7.59) we get:

(8.7) Then G$‘(Ic) can be interpreted as the normalization constant of the network with Ic jobs and node i removed from the network. G;‘(k)

=

c

fi

Fj(kj).

(8.8)

2 k.=K 27; i 3=1 3 &k,=K-k

By definition we have:

GN-~(k)=GE~(k)=GryN)(k)

for k=O,...,K.

Because of Eq. (8.7), for node N we have [Buze71, Buze73]: 7~(k) = #4iN-l(~

- IC).

(8.9)

316

ALGORITHMS FOR PRODUCT-FORM NETWORKS

In [BBS77], a simple algorithm for computing the G:‘(k) for k = , K is presented. Since the sum of all marginal probabilities is 1, 0 it’ follows from Eq. (8.7): &(j) j=o

= g

#Gf(K

- j) = 1,

and therefore we have: K G(K)

=

xF,(j)+G;)(K-j), j=o

j =

(8.10)

1,~..,N.

With the help of these equations we can derive an iterative formula for computing the G”)(k), for k = 0 7”‘? K** N

G;‘(k) = G(k) - &(j).G$(k j=l

- j),

(8.11)

with the initial condition: G$O)=G(O)=l

7 i=l

Y”‘? N *

(8.12)

In the case of rni = 1 the preceding formulae can be considerably simplified . Because of Eq. (8.11) we have: k-l

G:)(k)

= G(k) - c

Fi(j + l)G;‘(k

- 1 - j)

j=O = G(k)

k-l c

- ;

F,(j)G;‘(k

- 1 - j)

' j-0

= G(k) - ;G(k

- 1).

After inserting this result in Eq. (8.7), we get for the marginal probabilities:

7r#c) = (~)k.~.(G(K-~)-~,G(K-ii-l)),

(8.13)

where G(k) = 0 for k < 0. (b) The throughput of node i in the load-dependent or load-independent case is given by the formula:

X(K) = G(K - I) G(K)

and

Xi(K) = e;-

G(KG(K)

1) *

(8.14)

THE CONVOLUTION ALGORITHM

Proof:

By definition, the throughput

is given by Eq. (7.22):

= &&k

- /!Y)

- l)G$)(K

317

k=l K-l

= &

c

Fi(k)Gg’(K

- 1 - Ic) = ei ‘gi)l).

q.e.d,

k=O

(c) Th e u tl’z zxat ion of a node in the load-independent case can be determined by inserting Eq. (8.14) in the well known relation pi = Xi/(mipi): ei G(K-1) pi=------. mipui G(K)

*

(8.15)

(d) The mean number of jobs for a single server node can be computed from following simplified equation [Buze71]: (8.16)

(e) The mean response time of jobs at node i can be determined with the help of Little’s theorem. For a single server node we have: (8.17) The use of the convolution method for determining the normalization constant is now demonstrated by a simple example. We also show how to compute other performance measures from these results. Example 8.1 Consider the following closed queueing network (Fig. 8.3) with N = 3 nodes and K = 3 jobs. The first node has ml = 2 and the second node has m2 = 3 identical service stations. For the third node we have m3 = 1. The service time at each node is exponentially distributed with respective rates: ~1 = 0.8 set-‘,

p:! = 0.6 set-‘,

p3 = 0.4 set-1 ,

318

ALGORITHMS FOR PRODUCT-FORM NETWORKS

fig. 8.3

A closed queueing network.

The visit ratios are given as follows: el = 1,

e2 = 0.667,

e3 = 0.2.

From Eq. (7.61) it follows for the functions Fi(ki),

i = 1,2,3:

Fl(O)

= I,

F2(0)

F3(0)

= 1,

Fl(1)

= 1.25,

Fz(1) = 1.111,

F3(1)

= 0.5,

Fz(2) = 0.617, &(3) = 0.229,

F3(3)

Fi(2) = 0.781, Fi(3) = 0.488,

= 1,

Fa(2) = 0.25, = 0.125.

In Table 8.1 the intermediate results as well as the final result of computing G(K) are shown. Table 8.1 Computation ization constant

0 1 2 K=3

of the normal-

1

2

N=3

1 1.25 0.781 0.488

1 2.361 2.787 2.356

1 2.861 4.218 4.465

Thus the normalization constant G(K) = 4.465. The marginal probabilities for the single server node 3 can be computed using Eq. (8.13): =

= 0.528,

.iT3(1) =

= 0.312,

r3(0)

THE CONVOLUTION ALGORITHM

r,(2)

=

x3(3)

= (z)3.&.

(;)2&.

(G(1) - zG(O))

(G(O)- z.0)

319

=0.132, =0.028.

For the computation of the marginal probabilities of nodes 1 and 2, we need the values G$ (k) for k = 0, 1,2,3 and i = 1,2. With Eq. (8.11) we get: G(l)(O) = 1 G&l)

= G(1) - R’i(l)G$$)(O) = 1.611,

G:)(2)

= G(2)

- (Fi(l)Gg)(l)

+ &(2)G$(O))

G$)(3) = G(3) - (Fi(l)Gg)(2)

+ &‘i(2)Gg)(l)

= 1.423, + Fi(3)Gc)(O))

= 0.940.

In the same way we compute: GC2)(3) N = ** 1 316 GC2)(2) = -2.-.-T 175 N

G$)(O) = 1 7 Gg)( 1) = 1.656,

With Eq. (8.7) the marginal probabilities

Tl(“) = F(0)G(1)(3) G(3) N

are:

= L, 0 398

7rl(l) = -F1(1) @(2) G(3) N

0 211 -- i?

0 109 7ri(3) = -F1 (3) G(l) (0) = A, G(3) N

7rl(2) = -F1(2) G(l) (1) - 0 282 G(3) N - d’ and 740)

= 0.295,

The throughputs X1 = el g

n2(l) = 0.412,

x2.3) = 0.051.

can be computed using Eq. (8.14): = 0.945,

The utilizations

7r2(2) = 0.242,

X2 = e2g

= 0.630,

X3 = e,g

= 0.189.

are given by Eq. (7.21):

p1 = - x1 = 0.590,

x2

p2 = ___

ml/-J1

= 0.350,

p3

x3

= K = 0.473.

m2l-L2

The mean number of jobs at the multiserver nodes is given by Eq. (7.26): 771 = 7rl(l) + 27rr(2) + 37rr(3) = 1.290, 1T2 = 7r2(1) + 27r2(2) + 3n2(3) = 1.050, where we use Eq. (8.16) for the single server node 3: E3=(~)gg+(~)2~+(~)3~=o.660.

320

ALGORITHMS FOR PRODUCT-FORM NETWORKS

For the mean response time we use Eq. (7.43):

-

Kl = 1.366, T1 = x1 8.1.2

Multiclass

Closed

K2 = 1.667, T2 = x2

T3 = % x3 = 3.496.

Networks

The convolution method introduced in Section 81.1 can be extended to the case of multiclass closed product-form queueing networks. According to the BCMP theorem, Eq. (7.80), the state probabilities for closed product-form queueing networks are given by:

$31,. ..)S,) =& fpio, 2=1

where the functions Fi(Si) are defined in Eq. (7.82). The determination the normalization constant of the network:

of

G(K) = c fiE(si), is analogous to the single class case. For now, we assume that class switching is not allowed and thus the number of jobs in each class is constant. Corresponding to Eq. (8.2), we define for k = 0, . . . , K the auxiliary functions:

G(k)= c fi fi(si)

(8.18)

2 S,-ki’l i=l

and determine for n = 1, . . . , N the matrix G, by convolution: G, = F, @ Gn-1, with the initial condition Gr(.) = Fl(.).

For n > 1 we also have:

C&(k) = 5 . . . 2 F,(j).G+l(k j,=o j,=o

- j).

(8.19)

The computation of the normalization constant was simplified by [MuWo74b] and [Wong75] by introducing a much simpler iterative formula for G(K) = GN(K). For this purpose, we define the vector: kc”) =pSi i=l

= (Ici”‘,...,$)),

321

THE CONVOLUTION ALGORITHM

where kp) ,1 5 T < R, is th e overall number of jobs of class r at the nodes 1,2,..., n - 1, n. We then have: ky’ = 2

kir

and

kcN) = K.

i=l

With the help of these expressions, it is possible to rewrite Eq. (8.18) in the following manner:

G,(k(“))= C

fJFi(Si).

(8.20)

5 &=k(n) ‘=l i=l

Thus: n-l

G,,(k(“))

=

C

c k(‘+l)+S,=k(n)

n-1 C

n

Fi(Si)eFn(sn)

i=l

&=k(“-l)

i=l

= c

(8.21)

Gn-l(k(“-‘))*F,(S,).

k(n-1)+$&(n)

Equation (8.21) together with the initial conditions Gr(.) = J’r(.) completes the description of the algorithm of [MuWo74b] and [Wong75]. For the normalization constant we have: (8.22)

G(K) = G,v(kcN))e The computation of the performance measures is as follows:

(a) According to Eq. (7.32) th e marginal probability that there are exactly Si = k jobs at node i is given by:

r;(k)= c ~(%...&v) =& 2

c fiF,(S,) 5

Sj=K

Sj=K3=l

3=1

3=1

&S,=k

&Si=k

E(k) = Go

c ~ j=l

Sj=K~~~i

&Si=k

fi

Fj(Sj)

= $$Gg(K-k),

(8.23)

322

ALGORITHMS FOR PRODUCT-FORM NETWORKS

Then G:)(K) can again be interpreted as the normalization constant of the network without node i: (8.24)

In the same way as in the single class case (Eq. (8.11)), the iterative formula for computing the normalization constant G:‘(k) is given as follows: S(j). Fi(j). G$‘(k - j),

GE)(k) = G(k) - 5

(8.25)

j=O

with S(j) defined by: 0, ifj=O, 1, otherwise

Ki> =

(8.26)

and affecting the computation only for j = 0. The initial condition is: GE)(O) = G(0) = 17 i = l,...,N.

(8.27)

(b) The throughput Ai, of class-r jobs at the ith node can be expressed as follows [Wong75] :

where ei, is the solution of Eq. (7.72). With (K - lT) = (Kl, . . . , K, 17* * * , KR), we have the following simpler form: x

=e

GW-

ir

ir

14

G(K)

(8.28)

*

This formula holds for load-dependent and load-independent all four types. The proof can be found in [BrBa80].

nodes of

(c) The utilization pir is given by Eq. (7.34). If the service rates are constant, then we have the following simplification because of Xir/(mipir): pir

=

- Air miPui7-

=

-. eir miPir

G(K

G(K)

lr)

(8.29) *

THE CONVOLUTION ALGORiTHM

Fig. 8.4

323

An example of a two-class closed network.

Table 8.2 Computation F,(Si), i = 1,2,3,4,

W)

1

(LO) (W (171) (0,2>

1 4 4

(x4

12

of

1 1.6 2 6.4 4 19.2

‘2

the

functions

1 2.4 4.8

1 3.2 3 19.2 9 86.4

11.52 11.52 27.648

Example 8.2 Consider the network given in Fig. 8.4 consisting of N = 4 nodes and R = 2 job classes. In class 1 the number of jobs is Ki = 1 and in class 2, K2 = 2. It is assumed that class switching is not allowed. The first node is of Type-2, the second and third nodes are of Type-$, and the fourth node is of Type-3 with mean service times: 1

1 -=lsec,

1 -=4sec,

1 -=8sec,

-

Pll

CL21

I-L31

CL41

-

1

CL12

=2Sec,

-

1

= 5sec,

CL22

-

1

= lOsec,

CL32

-

1

= 12sec, = 16sec.

p42

The visit ratios are given by: ell = 1, e21 = 0.4, el2 = 1, e22 = 0.4,

e31 = 0.4, e32 = 0.3,

e41 = 0.2, e42 = 0.3.

First, with the help of Eq. (7.82), the functions Z?i(Si),i = 1,2,3,4, are computed [Table 8.2). Then for determining the normalization constant we compute the G,(kcn)) from Eq. (8.21) where Gr(.) = Fi(.). For n = 2 we

324

ALGORITHMS FOR PRODUCT-FORM NETWORKS

have:

Gz(O,0) = Gl(O,O)F2(07

=1 = iii,

0)

G2(l, 0) = G1(0,O)F2(1,0) + G~(l,o)Fz(o, 0) G2(0, 1) = Gi(0, O)F2(0, 1) + Gl(O, l)&(O, 0)

= 4,

Gz(1, 1) = G1(O,O)F2(Ll> + Gl(L l)F2(0,0> + G1(1,0)F2(0, 1) + Gi(O,l)F&O) G2(0,2) =

= 15.6,

+ Gl(O,2)J’2(07

G1(0,0)~2(0,2)

0) =l2,

+ Gl(O, l)J’2(0,1> G2(1,2) =

G1(0,0)5’2(17

+ Gl(l,

2)

2)F2(07

0)

+ G1(l, l)Fz(O, 1) + G1(0,1)F2(1,1) + Gl(l,O)F$I, 2) + Gl(0,2)F2(1,0)

= 62.4.

In the same way we can compute the values for Gs ( kc3)) and G4 (kc4)) as summarized in Table 8.3. So the normalization constant is G(K) = 854.424. Computation of G,(kci)) z

Table 8.3 ktn), 1 < n 5 N

G:!(kc2))

GS(kc3))

G4(kc4))

Kw)

1 2.6 4 15.6 12 62.4

1 5.8 7 55.4 33 334.2

1 8.2 11.8 111.56 78.12 854.424

(170) (071) (14

vv) W)

Now, with the help of Eq. (7.47), we can compute the marginal probability of 0 jobs of class 1 and 2 jobs of class 2 at node 4: 7r4(0,2)

F4(07‘) Gc4’(l 0) = L7 0 0782

= -

G(K)

N

’

where G$)(l , 0) can be obtained from Eq. (8.26): G;)(l,O)

= G(l,O) - F4(1,0).G$$(0,0)

Similarly other marginal probabilities r4(172)

n4(17

I>

=

F4(17 -G,

=

G(K)

2)

(4)

= 5.

at the nodes can be calculated:

(0,O) = 0.0324,

with Gc4)(0 0) = 1 N

F4(I’ ‘) Gc4)(0 1) = 0 0944 N

with Gc4)(0 7 N ’ 1) = -7

G(K)

7

L7

7

7

THE CONVOLUTION ALGORITHM

1) = 0.3112,

7r4(0,

‘)GC4)(0 2) = L, 0 0927 N ’

with GC4) 55 4 N (1 ’ 1) = L, with G(“)(O N ’ 2) = 33 ’

7r4(1)0)

= -F4(l,

n4(0,0)

F4(o,o)G(4)(l = m N

7r3(1,2)

=

-GN G(K)

~3(1,2)

(3)

(0,O) = 0.1011,

with GC3) N (0 ’ 0) = 1 ’

7r3(1,0)

=

-GN G(K)

F3(1,0)

(3)

(072) = 0.1600,

with GC3) 42 72 N (0 ’ 2) = L’

7r3(1,1)

=

-GN G(K)

F3(1,1>

(3)

(071)

with GC3)(0 88 N ’ 1) = A*

G(K)

’

2) = -L-----Y 0 3911

=

0.1977,

325

with GC4)(1 334 2 N ’ 2) = L,

The rest of the marginal probabilities are computed in the same way. The mean number of jobs can be determined with the help of Eq. (7.40). For example, for node 3 the mean number of class 1 and class 2 jobs is respectively given by: = 0.4588, K31 = ~3(1,0)+~3(1,1)+~3(1,2) 1?32 = r3(0,1) + x3(1,1) + 2r3(0, 2) + 2~3(l, 2) = 0.678. For the IS-node 4 the mean number of jobs by class type is: x41

K41 = -

= 0.219,

x42

z42 = -

= 0.627.

I-L42

P41

The throughputs at each node by class type can be computed with Eq. (8.28):

For the computation of the utilizations of each node by class type, Eq. (8.29) is used to get: p11 = 0.0914, p12 = 0.2611,

p21 = 0.1463, pz2 = 0.2611,

p31 = 0.2926, /I32 = 0.3917.

326

ALGORITHMS FOR PRODUCT-FORM NETWORKS

The algorithm we presented in the preceding text is applicable only to networks without class switching. For networks with class switching, [Munt72] proved that a closed queueing network with U ergodic chains is equivalent to a closed network with U job classes without class switching. Therefore, the computation of the normalization constant for networks with class switching is an extension of the technique for networks without class switching. More details are given in [BBS771 and [.BrBa80] and Section 7.3.6. In [ReKo75] and [Saue83] the convolution method has been extended for analyzing open queueing networks with load-dependent service rates and for analyzing closed queueing networks with load-dependent routing probabilities. The methods can also be extended for networks with class specific service rates. These are networks in which the service rates depend not only on the overall number of jobs of each class at the node, but also on the number of jobs at the node. For this case, [LaLi83] modified the convolution algorithm so as to make the best use of storage space. Their algorithm is known as tree-convolution. Because the computation of the normalization constant can cause numerical problems, other techniques were developed that allow the calculation of the performance measures without using the normalization constant. One of the key development in this regard is the mean value analysis (MVA), which we discuss next.

8.2

THE MEAN VALUE ANALYSIS

The MVA was developed by Reiser and Lavenberg [ReLa80] for the analysis of closed queueing networks with product-form solution. The advantage of this method is that the performance measures can be computed without explicitly computing the normalization constant. The method is based on two fundamental equations and it allows us to compute the mean values of measures of interest such as the mean waiting time, throughput, and the mean number of jobs at each node. In these computations only mean values are computed (hence the name). For the case of multiserver nodes (mi > 1)) it is necessary, however, to compute the marginal probabilities. The MVA method is based on two simple laws: 1. Little’s theorem, which is introduced in Eq. (6.9) to express a relation between the mean number of jobs, the throughput, and the mean response time of a node or the overall system: K=X.T. (8.30) 2. Theorem of the distribution at arrival time (in short, arrival theorem), proven by [LaRe80] and [SeMi81], for all networks that have a productform solution. The arrival theorem says that in a closed product-form queueing network, the pmf of the number of jobs seen at the time of arrival to a node i when there are Ic jobs in the network is equal to the

THE MEAN VALUE ANALYSIS

327

pmf of the number of jobs at this node with one less job in the network (= (k - 1)). Th is property has an intuitive justification [LZGS84]. At the moment a job arrives at a node, it is certain that this job itself is not already in the queue of this node. Thus, there are only Ic - 1 other jobs that could possibly interfere with the new arrival. The number of these at the node is simply the number there when only those (K - 1) jobs are in the network. At first we introduce the MVA for single class closed queueing networks and explain it in more detail. This algorithm is then extended to multiclass networks, mixed networks, and networks with load-dependent service rates. 8.2.1

Single

Class Closed

Networks

The fundamental equation of the mean value analysis is based on the arrival theorem for closed product-form networks [ReLa80, SeMi81] and it relates the mean response time of a job at the ith node and the mean number of jobs at that node with one job less in the network, that is: Ti(K)

= 2-m [I +QK - I)] , i = l,...,N. (8.31) Pi For single server stations (mi = 1) with an FCFS strategy, it is easy to give an intuitive explanation for Eq. (8.31) because for each FCFS node i the mean response time Ti(.K) of a job in a network with K jobs is given by the mean service time (l/pi) of that job plus the sum of the mean service times of all jobs that are ahead of this job in the queue. Equation (8.31) can also be derived without using the arrival theorem. For this purposes, we use the formulae for computing the utilization (Eq. (8.15)) and the mean number of jobs (Eq. (8.16)): dK)

ei G(K - 1) = E’ G(K) 7

and:

17i(K) =2 (~)k*G&? k=l

From Eq. (8.16) it follows:

If we transform Eq. (8.15) for G(K-1) and insert the result of the preceding equation, then, after rearranging the equation, we get: pi(K)*?7i(K

- 1) = 5 k=2

(E)”

* G$i)li)

7

ALGORITHMS FOR PRODUCT-FORM NETWORKS

328

and substituting Ki(K)

this equation to Eq. (8.16), we have:

G(K - 1) + Pi(h’)Z(K - 1) (-qK) = 1;' = Pik) + Pi(K)-&(K - 1) = pi(K)ei

[l + K;(K

- l)] .

If we assume constant service rates and rni = 1, then we get the desired result by using pi(K) = Xi(K)/pi and Little’s theorem: -

T;(K)

Ki(K) = Xi(K)

pi(K) = -Xi(K)

- 1)] = 1. [l + Ki(K

[l +Ki(K

- l)] .

Pi

For computing the mean response time at the ith node we need two othLer equations in addition to Eq. (8.31) to describe the MVA completely. Both equations can be derived from Little’s theorem. The first one is for determining the overall throughput of the network: (8.32)

whereas the other one determines the mean number of jobs at the ith node: Ki(k) = X(k).Ti(k). ei, (8.33) where ei is the visit ratio at the ith node. The three equations, (8.31), (8.32), and (8.33), allow an iterative computation of the mean response time, mean number of jobs, and throughput of the closed product-form queueing network. The iteration is done over the number of jobs Ic in the network. Equation (8.31) is valid for FCFS single server nodes, PS-nodes, and LCFS PR-nodes. The description of the MVA is complete if we extend Eq. (8.31) to the case of IS-nodes and FCFS-nodes with multiple servers. In the case of IS-nodes we have: Ti(K)

= ;.

i

(8.34)

For the latter case, consider a job that arrives at -/M/m-FCFS node containing j - 1 jobs, given that the network population is Ic - 1. This event occurs with probability 7ri(j - 1 1Ic - 1). Then we have:

T&k)=k

3

jE1 Pi * %(d

&i(j) =

- 7ri(j - 1 1k - l),

if j < mi, 3, 1.m,i , otherwise.

329

THE MEAN VALUE ANALYSIS

To obtain an expression for the probability ni(j ( F;), we use the formulae presented in the convolution algorithm for computing the performance measures with help of the normalization constant, namely:

ni(jp)=Fi(j)*

G’~~k~ ‘)*

(see Eq. (8.7)) G;‘((k

ei = pi . &)

- 1) - (j - 1))

* Fi(j - 1) -

1

W)

= pi * w(j)

ei - G@ - 1) . G:)(@ - 1) - (j - 1)) . qj _ l) ‘L G(k - 1) , *, wv / L v ri(j--ilk-1) see Eq. (8.7) h(k)

W) = pi *&i(j)

* %(j - 1 ( k - l),

By induction and rearrangement of the equations, the following can be obtained:

=-’

1 mi f Pi

rni -2

1+K&-1)+ (

C(mi j=o

-

-

1)-%(jIk- 1) , (8.35)

=I--.

%(j I q =

1 mi

b@) Pi ’ w(j)

77lj-j)

*ri(j 1k)

queueing networks can

For i = 1,. . . , iV and j = 1,. . . , (mi - 1):

Xi(O) = 0,

Ti(O 10) = 1,

(8.36) (8.37)

- 7ri(j - 1 1 Ic - 1).

Now the MVA for closed single class product-form be described as follows: Initialization.

7

Xi(j I 0) = 0.

330

ALGORITHMS FOR PRODUCT-FORM NETWORKS

Iteration over the number of jobs k = 1,. . . , K. L.STEP 2 I Fori=l,... the ith node:

, N, compute the mean response time of a job at

; a[I l tKi(k- l)] , T&q = A-

l +l?$c

L 1

mi-2

- 1) +c

[

Pi**i

Type-1,2,4 (*i = I>,

(rni - j - l)*%(j

I Ic - 1)

1

Type- 1

’ (*i

j=o

>

I),

Type-37

ii’

(8.38)

where the conditional probabilities are computed using Eqs. (8.37) and (8.36). [STEP]

Compute the overall throughput:

X(k)= N Ic c ei.T&

(8.39)

i=l

The throughput

of each node can be computed using the following equation: Xi(k) = ei* X(k),

(8.40)

where the ei can be determined with Eq. (7.5). I STEP 2 3 node:

For i = 1, . . . , N, compute the mean number of jobs at the ith xi(k)

= ei. X(k)‘Ti(k)*

(8.41)

The other performance measures, e.g., utilization, mean waiting time, mean queue length, etc., can be derived from the calculated measures using the wellknown equations. The disadvantage of the MVA is its extremely high memory requirement. The memory requirement can be considerably reduced by using approximation techniques (e.g., SCAT, or self-correcting approximation technique), discussed in Chapter 9. Another disadvantage of the MVA is the fact that it is not possible to compute state probabilities. In [AkBo83], the MVA has been extended for computing the normalization constant and for computing Only Step 2.2 needs to be extended to include the the state probabilities. equation: G(k) =

G(k - 1) qjq 7

(8.42)

with the initial condition G(0) = 1. This follows immediately from Eq. (8.14). When the iteration stops, we have the normalization constant G(K) that

THE MEAN VALUE ANALYSIS

331

can be used to compute the state probabilities with the help of the BCMP theorem, Eq. (7.80). T wo applications of the mean value analysis are now given. Example 8.3 The central-server model shown in Fig. 8.5 has N = 4 nodes and K = 6 jobs. The service time of a job at the ith node, i = 1, 2, 3, 4, is exponentially distributed with the following mean values: 1 1 - = O.O2sec, - = 0.2sec, P2 I-L1

-

1

-

= 0.4sec,

P3

1

= 0.6sec.

CL4

-cYutGL,H3PI )lIIn-@-clYIE+ Fig. 8.5 The central-server model. The visit ratios are given as follows: el = 1,

e2 = 0.4,

e3 = 0.2,

e4 = 0.1.

With the help of the MVA, the performance measures and normalization constant of the network are computed in three steps: Initialization: x1(0) = K2(0) = K3(0) = z4(0) = 0, Iteration m

over the number of jobs in the network starting with

Mean response times, Eq. (8.38): E(1) T3(1)=

1-1

G(0) = 1.

= ; ;

[l -t%(o)]

= 0.02,

T2(1) = ;

[l +Kz(O)]

= 0.2,

[l+K3(0)]

=0.4,

T4(1)

[l+K4(0)]

=oJ%

Throughput, X(l)=

*

= ;

Eq. (8.39)) and normalization constant, Eq. (8.42): l

C &(l) i=l

= 4.167,

G(1) = #

= 0.24.

332

ALGORITHMS FOR PRODUCT-FORM NETWORKS

Tab/e 8.4 steps.

Performance measures after completing six iteration

Node

1

2

3

4

Mean response time !Fi

0.025

0.570

1.140

1.244

Throughput Xi Mean number of jobs K, Utilization pz

9.920 0.244

3.968 2.261

1.984 2.261

0.992 1.234

0.198

0.794

0.794

0.595

-1

Mean number of jobs, Eq. (8.41): El(l)

= X(l)Ti(l)ei

= 0.083,

Kz(l)

= X(l)Tz(l)ez

E,(l)

= X(l)Ts(l)es

= 0.333,

F4(1) = X(l)Td(l)e4

= 0.333, = 0.25.

Iteration for k = 2: -1

Mean response times: Tl(a> = ;

[l +%(I)]

= 0.022,

Tz(2) = ;

[l +K2(l)]

= 0.267,

573(2>

[l + K3(1>]

= 0.533,

T4(2) = i

[l +K4(1)]

= 0.75.

=

-1

;

Throughput X(2) =

4

and normalization

2

= 6.452,

constant:

G(2) = z

= 3.72. 10-2.

C eiTi(2) i=l

STEP 2 3 Mean number of jobs: l

Ki(2)

= X(2)?;1(2)el = 0.140,

1?2(2) = A(2)T2(2)e2 = 0.688,

z,(2)

= X(2)F3(2)e3 = 0.688,

K4(2) = A(2)T*(2)e4 = 0.484.

After six steps, the iteration stops and we get the final results as summarized in the Table 8.4. The normalization constant of the network is G(K) = 5.756 +lo-“. With Eq. (7.80) we can compute the steady-sate probability that the network is, for example, in the state (3,1,1,1): h

=

5.337.10-4.

THE MEAN VALUE ANALYSIS

Fig. 8.6

333

A closed queueing network.

Example 8.4 As anot her example, consider the closed queueing network given in Fig. 8.6 with K = 3 jobs. At the first node we have ml = 2 identical processors having exponentially distributed service times with mean l/,~r = 0.5 sec. Node 2 and node 3 have exponentially distributed service times with means l/p2 = 0.6 set and l/bs = 0.8 set, respectively. At the fourth node (terminals), the mean service time is l/p* = 1 sec. The routing probabilities are as follows: P12

=

~13

= 0.5

and

~24 = p34 = p41 = 1.

From Eq. (7.5) we compute the visit ratios: el = 1,

e2 = 0.5,

e3 = 0.5,

e4 = 1.

The analysis of the network is carried outin the following steps: ~~~

Initialization: mo)

= K2(0)

Iteration

I-2.1

IsTEP

= 0,

x1(0 IO) = 1,

7rl(l

IO) = 0.

over the number of jobs in the network starting with

Mean response times:

Ti(1) = T3(1)

= K3(0)

&

= i

[1+ [1+

K(0)

K3(0)]

Throughput:

+ v(o = 0.8,

1 O)] = o.s,

Tz(1)

= ;

[l + F2(0)]

T4(1)

= ;

=A.

= 0.6,

334

ALGORITHMS FOR PRODUCT-FORM NETWORKS

-1

Mean number of jobs: El(l)

= ~(l)~l(l)el

= 0.227,

z,(l)

= X(l)Tz(l)ez

= 0.136,

Es(l)

= X(l)Fs(l)es

= 0.182,

??~(l) = X(l)FA(l)ed

= 0.454.

Iteration for k = 2: -1

Mean response times: 532)

=&

[1+~1(1)+m(o1l)] =0.5,

with: 7rl(O 11) = 1- $

[

1

:x(l)

+7r1(1 1 1) = 0.773,

and:

d1 1l> = ~~(l)n,(o 10) = 0.227,

T2(2)

%(2>= ; [I t-%(l)] = 0.946, -1

= ; [I+??$)] = 0.682,

T4(2) = 1.

Throughput: x(2) =

= 0.864. 4 2 T f?;?;;(2) u -6

i=l

-1

Mean number of jobs: K1(2)

=

X(2)T1(2)el

0.432,

Kz(2)

= A(2)Tz(2)ez = 0.295,

=

X(2)T3(2)e3 = 0.409,

F*(2)

= X(2)?;4(2)ed = 0.864.

=

-

K3(2)

Iteration for k = 3: -1

Mean response times: C(3)

= f-& [I +x1(2)+

nl(0 I2)] = 0.512,

with:

1

= 0.617,

THE MEAN VALUE ANALYSIS

335

and:

w(l

l’sTEf2.23

12)

=

32)711(0

1 1) = 0.334,

%(3) = ;

[l -t-&(2)]

= 0.776,

T3(3) = i

[l +??,(a)]

= 1.127,

Ej(3) = 1.

Throughput: X(3) =

/?!?%~~

= 1.217.

3 5 &(3) i=l

Mean number of jobs:

E,(3)

= X(3)T1(3)el = 0.624,

17,(3) = X(3)&(3)e2

KS(~) = X(3)T3(3)e3 = O.686! z*(3)

= 0.473,

= X(3)F4(3)e4 = 1.217.

The throughput at each node can be computed with Eq. (8.40): X1 = X(3). el = 1.218, X3 = X(3). e3 = 0.609, For determining the utilization

p1=

8.2.2

h

~ m1p1

Multiclass

= 0.304,

Closed

X2 = X(3). e2 = 0.609, X4 = X(3). e4 = 1.218.

of each node, we use Eq. (7.21): P2

x2

= z

= 0.365,

x3

p3 = -

P3

= ().487. -

Networks

The algorithm for computing the performance measures of single class closed queueing networks can easily be extended to the multiclass case in the following way [ReLa80] : Initialization.

For i = 1,. . . , N, T = 1,. . . , R, j = 1,. . . , (mi - 1):

J&(O,O...,O) =o, 7ri(O IO)= 1, %(j IO>=a IteraCon: k = 0, . . . , K:

336

ALGORITHMS FOR PRODUCT-FORM NETWORKS

[ STEP 2.1 1 For i = 1, . . . , N and r = 1, . . . , R, compute the mean response time of class-r jobs at the ith node:

’

&

. [

1+ e Eis(k - 1,) s=l

1 Fir(k)

1

Type-1,2,4 (mi = I>,

1+ F Fis(k - 1,)

= ( /-hr. mi [

s=l

(8.43)

m,-2

-t- c (m; - j - l)r&’ j=o

] k - 1,)

1 -3

1

, Type-1 (mi > l), Type-3.

x Pir

Here (k - 1,) = (ICI, . . . , Ic, - 1, . . . , /CR) is the population vector with one class-r job less in the system. The probability that there are j jobs at the ith node (j = 1,. . . , (mi - 1)) given that the network is in state k is given by: ri(j

1k)=

h 2

"i'Xr(k)ri(j-

3

Pir

r=l

11 k-

1,)

1 ,

(8.44)

and for j = 0 by: ?ri(O]k)=l-;

z

5

%X,(k)

+ mgl(mi

r=l

- j)r&’

j=l

] k)

1

(8.45)

where ei, can be computed by Eq. (7.72). 1 STEP 2.2 ]Forr=l,... , R, compute the throughput: X,(k) = N C

(8.46)

Icr *

eirTir(k)

i=l

1_______1

STEP 2 3 For i = 1, . . . , N and r = 1, . . . , R, compute the mean number of class-r jobs at the ith node: Fir(k)

= Xr(k)*Tir(k)*

ei,.

(8.47)

With these extensions the MVA algorithm for multiclass closed product-form queueing networks is completely specified. Akyildiz and Belch [AkBo83] extended the MVA for computing the normalization constant and the state probabilities. For this purpose, Step 2.2 in the preceding algorithm is expanded to include the formula:

G(k) =

G(k - 1,) L-(k)

’

(8.48)

THE MEAN VALUE ANALYSIS

337

with the initial condition G(0) = 1. After the iteration stops we get the normalization constant G(K) that can be used to compute the steady-state probabilities with the help of the BCMP theorem, Eq. (7.80).

Fig. 8.7

Sequenceof intermediate values.

To explain iteration Step 2, we use Fig. 8.7 where the sequence of intermediate values is shown for a network with R = 2 job classes. The first job class has K1 = 2 jobs and the second class has K2 = 3 jobs. The mean value algorithm starts with the trivial case where no job is in the system, that is, the population vector (0,O). From this, solutions for all population vectors that consist of exactly one job are computed; in our example these are the population vectors (1,0) and (0,l). Then the solution for all population vectors with exactly two jobs are computed, and so on, until the final result for K = (2,3) is reached. In general, to compute the solution for a population vector k we need R intermediate solutions as input, namely the solutions for all population vectors k - l,, r = 1,. . . , R. Now, the algorithm is illustrated by means of an example: Consider the queueing network shown in Fig. 8.8 with N = Example 8.5 3 nodes and R = 2 job classes. Class 1 contains K1 = 2 jobs and class 2 contains Kz = 1 jobs. Class switching of the jobs is not allowed. The service time at the ith, i = 1, 2, 3, node is exponentially distributed with mean values: 1 = 0.2sec, Pll 1 -=0.2sec, Pl2

-

1

= 0.4sec,

P21

-

1

i-422

-

1

= 1 set,

I-431

=0.6sec,

-

1

p32

=2sec.

338

ALGORITHMS FOR PRODUCT-FORM NETWORKS

Another closed queueing network.

Fig. 8.8

The visit ratios are given as follows: ell = 1,

e31 = 0.4,

e21 = 0.6,

= 1,

el2

e32 = 0.7.

e22 = 0.3,

The queueing discipline at node 1 is FCFS and at node 2, processor sharing. The terminal station is modeled as an IS-node. We analyze the network using the MVA in the following three steps: Initialization: Xi,(O)

= 0 f or i = 1,2,3 and T = 1,2,

7r1(0 10) = 1,

x1(1 10) = 0.

Iterate over the number of jobs in the network beginning with the population vector k = (1,O): -1

Mean response times, Eq. (8.43):

%(l,O) = & T&O)

= ;

T31(1,0) = ; -1

[I

+

%l(O,

0)

[1+~21(0,0)

+

%z(O,O>

+~22(0,0)]

+

m(0

I 0, o,]

= 0.4,

=I.

Throughput

h(U)

for class 1 using Eq. (8.46):

= 3 C eiA

= 1.190.

l (LO>

i=l

Note that the class 2 throughput for this population vector is 0.

=

0.2,

THE MEAN VALUE ANALYSIS

-1

Mean number of jobs, Eq. (8.47): E,,(i,O)

= X1(l,O)T&O)ell

= 0.238,

~,,(i,O)

=

= 0.286,

X1(l,0)~21(l,0)e2~

E31(1, 0) = X1(1, O)T31(1,O)e31 = 0.476. Iteration for k = (0,l): [I

Mean response time: ~12(0,1>

=

&

T22(0,1)

=

&

[I + %(o,

.

[l + 17&O, 0) +

?;32(0,1) = ; /?%$-%%I

0) + 1712(0,0)

E22(0,

o)]

+ m(0

I 0, O)]

= 0.2,

= 0.6,

= 2.

Throughput

for class 2:

X2(0,1)

=

3

C

= 0.562.

-l

1)

ei2Ti2(O:

i=l

Note that the class 1 throughput (-1

for this population vector is 0.

Mean number of jobs:

E,,(O,l)

= X2(0, 1)552(0,l)e12

= 0.112,

F22(0,i) = X~(O,l)T22(0,Q322 = 0.101, ES2(0, 1) = X2(0,1)5732(0,1)e32= 0.787. Iteration for k = (1,l): I-2.1

Mean response times:

Class 1:

T11(1,1) =&

[I + %2(0,1)

+ 7r1(0 10, l)] = 0.2,

where: n1(0 ) 0,l) = 1 - - l

%,(o,

ml

[ p12

1) + Tl(1 10,l)

1

= 0.888,

339

340

ALGORITHMS FOR PRODUCT-FORM NETWORKS

m(l I071) =

%,(O, CL12

1)7Tl(O ( 0,O) = 0.112,

T&,

1) = k

[I +2721(0,1)+~22@,1)]

%(l,

1) = &

= 1.

= 0.44,

Class 2: F&,1)

= -A2. P.12

[1+1711(1,0)

~~(1 1 I, 0) = $X1(1,O)n1(0

-1

( l,O)] = 0.2,

+ ~~(1 1 1,0)

E&(1,0)

~(0 ( l,o) = 1 - $

T22(1, 1) = k

+w(o

1

= 0.762,

( 0,O) = 0.238,

[l + K&O)]

= 0.772,

5%2(1,1)

Throughputs: =

J&,1>

3

C eil%(L

i=l X2(1,1>

= 1.157,

l

1) = 0.546.

1

=

5 ei2R2 (1, 1)

i=l

-1

Mean number of jobs: K&,

1) = X1(1, l)Tll(l,

l)ell

= 0.231,

K12(1, 1) = X2(1,1)T12(1,l)el2

= 0.109,

F&(1,1)

= 0.305,

= Xl(l,l)T21(1,l)e21

?f22(1, 1) = X2(1,1)T22(1,+22

= 0.126,

F&(1,1)

= !LB&

K32(I,

= X1(1,l)T31(l,l)e3l I) = X2(1, I)T32(1,l)e32

= O-764.

= h

= 2.

THE MEAN VALUE ANALYSIS

Iteration for k = (2,0): -1

Mean response times:

=&

%(2,0>

[1+~11(1,0)+n(O 1LO)] =0.2,

Tz1(2,0) = k

[I + &(l,

T31(2,0) = A

= 1.

1-1

O)] = 0.514,

Throughput: 2 WV)

= 2.201.

=

5 dil(2,O) i=l

p?%?$3

Mean number of jobs: S&(2,0)

= X1(2,O)T&

O)ell = 0.440,

xz1(2, 0) = &(2,O)Tz1(2,O)ezl

= 0.679,

KS1(2, 0) = X1(2, 0)‘?‘,1(2,O)e31 = 0.881. Iteration for k = (2,l): -1

Mean response times:

Class 1: [1+ K&,

1) + -T(12(1!1) +

n(O

I 1, I)]

=

0.203,

where: 7Tl(O ) 1,1) = 1 - -L %l(l,l) ml [ PII

+ %X&,1)

+ 7rl(l 11,l)

1

= 0.685,

and: v(l

I 171)

5%(2,1>

=

y$‘(l,

=

;

1)7rl(O 10,l) + EX?(l, [l +E&

1) +Ezz(l,

l)*m(O

I LO)

= 0.289,

I)] =

Class 2: T12(2,1>

=

&

[l +E,1(2,0)

+ 7rl(O 12,0)] = 0.205,

341

342

ALGORITHMS FOR PRODUCT-FORM NETWORKS

where: 7r1(0 ) 2,O) = 1 - -

%X1(2,0)

+ ~~(1 12,0)

1

= 0.612,

and: m(1

I w>

%(2,1) -1

=

~&(2,0)7rl(O

= &

) 1,o) = 0.335,

[1+ Kar(2,0)]

= 1.008,

Ts2(2,1) = &

= 2.

Throughputs: = 2.113, C d%

(2,1>

i=l

-1

i=l

Mean number of jobs: &(2,1)

= X1(2, 1)5711(2,l)ell

= 0.428,

E12(2, 1) = X2(2, l)T12(2,l)el2

= 0.108,

X2r(2, 1) = X1(2, l)T21(2,+21

= 0.726,

K22(2, 1) = X2(2, 1)5722(2,l)e22

= 0.158,

X731(2,1) =

0.8457

A1(2,l)T3i(:qu&eing

??,,(a, 1) = X2(2, l)T32(2,+32

= 0.734.

We can see that the MVA is easy to implement and is very fast. For networks with multiple server nodes (mi > 1) and several job classes, this technique has some disadvantages such as stability problems, accumulation of roundoff errors, and large storage requirements. The storage requirement is 0 (N. n,“=, (K, + 1)). For comparison, the storage requirement of the convolution algorithm is 0 (n,“=,(Kr cases is approximately

+ 1)). The time requirement in both

the same: 2 . R(N - 1). fi (K, + 1) [CoGe86]. Recall r=l

that N is the number of nodes in the network and K, is the number of jobs of class T. 8.2.3

Mixed

Networks

In [ZaWo81] the MVA is extended to the analysis of mixed product-form queueing networks. But the networks are restricted to consist only of single server nodes. Before we introduce the MVA for mixed queueing networks,

THE MEAN VALUE ANALYSIS

343

we consider the arrival theorem for open queueing networks. This theorem says that the probability that a job entering node i will find the network in state (ICI.. . . , Ici, . . . , ICN) is equal to the steady-state probability for this state. This theorem is also called PASTA theorem. r Therefore we have for the mean response time of a job in an open network:

(8.49)

As from [ZaWo81], we have, for any class r and s, Fi,

= PisKi,

and,

Pir

therefore, with the help of the relation Fir(k) Eq. (8.49):

= Xir*Tir(k),

Pir

pir

?Tir (k) =

1 -

5

it follows from

Type-1,2,4 (mi = l), Pis’

s=l

Pir 7

Type-3,

(8.50) with pir = Xir/pir, where Xi, = &.a ei, is given as an input parameter. These results for open queueing networks are now used for analyzing mixed queueing networks. The arrival theorem for mixed product-form queueing networks says that jobs of the open job classes when arriving at a node see the number of jobs in equilibrium, while jobs of the closed job classes when arriving at that node will see the number of jobs at that node in equilibrium with one job less in its own job class in the network [ReLa80, SeMi81]. If we index the open job classes with op = 1,. . . , OP and the closed job classes with cl = 1, . . . , CL, then we have for the mean number of jobs in an open class T, taking into account Eq. (8.50): Fir(k)

= Air* -L

1+ 5

Pir

Ki,cl(k)

+ E op=l

cl=1

Xi,,,(k)

1 (8.51)

I

= 1 -

E

>

Pi,op

op=l

‘PASTA:

Poisson

arrivals

see time

averages

(see [Wolf82]).

344

ALGORITHMS FOR PRODUCT-FORM NETWORKS

where k is the population vector of the closed job classes. Equation (8.51) is valid for Type-1,2,4 single server nodes, while for Type-3 nodes we have:

&r(k)

= pir.

(8.52)

For the mean response time of a job in a closed class r at node i we have: OP

R, .(k)

= L

1+

Pir

E

Ki,cl(k

-

1,)

c

+

Pir

-

1 CL

1 +

1+

Ki,,,(k

op=l

cl=1

r

=I

1 I,)

C

1 Ki,cl(k

-

cl=1 5

Ki,cl(k

-

lr)

+

1,)

OP C

1 op=

pi,op 1

OP

1- c Pi,op

cl=1

op=l

=-

11 1L1+

5

Ki,cl

(k

-

L-)

cl=1

1

A

(8.53)

j%r

1- E

Pi,op

op=l

Equation (8.53) is valid for Type-1,2,4 single server nodes. For Type-3 nodes we have: Fir(k)

=

(8.54)

$

With these formulae the MVA for mixed product-form queueing networks can be described completely. We assume for now that all arrival and service rates are load independent. The algorithm is as follows: Initialization. For all nodes i = 1,. . . , N, compute the utilization of the open class jobs op = 1,. . . , OP in the mixed network: 1 Pi,op

=

-X0,*

ei,0p,

(8.55)

Pi,op

and check for the ergodicity condition (pi,OP 5 1). Set Ki,cl(O) = 0 for all i= I,..., N and all closed classes CL= 1, . . . , CL. Construct a closed queueing network that contains only the jobs of the closed job classes and solve the model with the extended version of the MVA, which considers also the influence of jobs of the open job classes. The results are the performance measures for the closed job classes of the mixed queueing network.

THE MEAN VALUE ANALYSIS

Iteration:

345

k = 0,. . . , K:

I

STEP 2 I For i = 1,. . . , N and r = 1,. . . , CL, compute the mean response times with Eqs. (8.53) and (8.54). CL, compute the throughputs with Eq. (8.46). I STEP 2 2 Forr=l,..., STEP 2 3 For i = 1,. . . , N and T = 1,. . . , CL, evaluate the mean number of jobs with Eq. (8.47).

I

With the help of the given in Section 7.2, compute classes of the network starting mean number of jobs Ei,, i =

solutions of the closed model and the equations the performance measures for the open job with Eqs. (8.51) and (8.52) for computing the 1,. . . , N and r = 1,. . . , OP.

If we use the iterative formula (8.48) f or computing the normalization constant in Step 2, then the steady-state probabilities can be derived from the BCMP theorem, Eq. (7.76) [AkBo83]: (8.56)

n(S = Sl . . . ) S,) = where X(j) is the (possibly state dependent) arrival rate.

As an example of a mixed queueing network, consider the Example 8.6 model given in Fig. 8.9 with N = 2 nodes and R = 4 job classes. Class 1 and 2 are open, the classes 3 and 4 are closed. Node 1 is of Type-2 and node 2 is of Type-4. _---

Source

-

Open Class Closed Class

LJ

Fig. 8.9

A mixed queueing network.

The mean service times are given as follows: -

1

= 0.4sec,

Pll

-

1

P21

= 0.6sec,

1 = 0.8sec, CL12 1 = 1.6sec, CL22

-

1

-

1

p23

1

= 0.3seq

-

= 0.5sec,

1 = 0.8sec. k24

I-L13

P14

= 0.5sec,

346

ALGORITHMS FOR PRODUCT-FORM NETWORKS

The arrival rates by class for the open class jobs are Xi = 0.5 jobs/ set and X2 = 0.25 jobs/ sec. In each of the closed classes there is K3 = K4 = 1 job. The routing probabilities are: Pll,ll

= 0,

1,

p12,12

= 0,

PO,13 = 0,

P13,13

=

PO,14 = 0,

p14,14

= 0.6,

1,

PO,11 = PO,12 =

0.5,

0.5,

Pll,Zl

=

~12,22

= 0.6,

p13,23

=

pl4,24

= 0.4,

0.5,

P21,ll

=

1,

~22~2

=

1,

p23,13=

1,

=

1.

p24,l4

Thus, by Eq. (6.27) the visit ratios are computed to be: 2, e12 = 2.5, = 1, e22 = 1.5,

ell

=

e21

e13 = 1, e23 = 0.5,

el4 e24

= 1, = 0.4.

The determination of the performance measures of this mixed queueing network is carried out with the MVA in the following three steps: Initialization. job classes:

Compute the utilization

of both nodes by the open

PII = hell+ -!- = 0.4, l-L11

p21 = Xle21. L

~12 = X2el2.

~22 = X2e22-

2-

Pi2

= 0.5,

=0.3,

IQ21 k

=0.6.

Set Kii (0) = Xi2 (0) = X21(Q) = K22(Q) = 0. Using MVA, analyze the closed queueing network model, obtained by leaving out the jobs of the open job classes. Mean value iteration for k = (1,O): Mean response times, Eq. (8.53): 1+ Pl3

Throughput,

1 -

&3(Q) (Pll

= 3,

T23(1,0) = J-

+ P12)

CL23

1 + z,,(a) 1 -

Eq. (8.46) : X3(1,0) =

2 C

K3

= 0.182.

ei3%3(1,0>

i=l

Mean number of jobs, Eq. (8.47): K13(1,0)

= ~3(1,0)*

ei3.T13(1,0)

z23(1,0)

= ~3(1~o)’

e23- T23

= 0.545,

(1, 0) = 0.454.

(p21

+ ,022)

= 5’

THE MEAN VALUE ANALYSIS

347

Mean value iteration for k = (0,l):

%4(0,1> = 5,

T24(0,1> = s,

X4(0,1) = 0.122,

K14(0, 1) = 0.610,

&(O,

1) = 0.390.

Mean value iteration for k = (1,1):

T&(1,1)

= 4.829, T2&,1)

X3(1, 1) = 0.120, K13(l,1)

= 6.951, ?;I&,

1) = 7.727, %(l,

1) = 11.636.

1) = 0.624, %,(l,

1) = 0.376.

X4(1, 1) = 0.081.

= 0.582, &(l,

1) = 0.418, &(l,

With these results and using Eq. (8.51), the performance measures of the open classes can be computed: F1&

1> =

Pll

(1+

%3(L

1) + %4(L

1))

,

0.1 K21(1 1> = Pzl (1 + E23(1~~> + %dl, , 0.1 K12(1

1) =

P12 (1 + %3(1,1>

+ %4(L

1> =

p22 (1 + rc,,&

+ K24(1,1>>

7 K

(1 22

7

0.1

l>>

1))

= 8.822, = 5.383, = 11.028, = 10.766.

For the computation of the mean response times we use, Eq. (7.43): Tll (1,1) = 8.822,

T12(l, 1) = 17.645,

T21(1, 1) = 10.766,

T22(1, 1) = 28.710.

Now all the other performance measures can be computed. There are some suggestions to extend this algorithm to mixed queueing networks that are allowed to contain additional multiple server nodes. The most important approaches can be found in [KTK81] and [BBA84]. 8.2.4

Networks

with

Load-Dependent

Service

The MVA can be extended to examine product-form queueing networks where the service rates at node depends on the number of jobs at that node, [Reis81]. Let pi(j) denote the load-dependent service rate of the ith node when there are j jobs in it.

348

ALGORITHMS FOR PRODUCT-FORM NETWORKS

8.2.4.1 Closed Networks The algorithm for single class queueing networks with nodes having load-dependent service rates can be described in the following steps: Initialization. Iteration:

For i = 1, . . . , N:

Ic = 1, . . . , K.

1 STEP 2.1 ] For i=

l,...

, N, compute the mean response time:

Ti(lc)= & -7ri(j 3 j=l l-G>

-1

7ri(O 10) = 1.

(8.57)

- 1 1Ic - 1).

Compute the throughput:

X(k)= N Ic . c ei.T,(k)

(8.58)

i=l

I STEP 2

3 Fori=l,..., N, compute the conditional probabilities of j jobs at the ith node given that there are Ic jobs in the network:

W)

-ri(j cLi(j) ri(j

I k) =

- 1 ) k - l)ei,

for j = 1,. . . , Ic, (8.59)

I

for j = 0.

The extension of this algorithm to multiclass queueing networks where service rate of a node depends on the overall number of jobs at the node be done very easily. Let hir(j) be the the service rate of class-r jobs at ith node when there are j jobs in it. Then the MVA can be presented in following steps: Initialization. Iteration:

For i = 1, . . . , N:

the can the the

7ri(O I 0) = 1

k = 0,. . . , K.

[ STEP 2.1 ] For i = 1, . . . , N and r = 1, . . . , R, compute the mean response times: Tir(k)

= 2 j j=l f4.d

T;(j--Ilk-l,),

where k=e&. r=l

(8.60)

THE MEAN VALUE ANALYSIS

I STEP 2

2 Forr-

349

, R, compute the throughputs:

l,...

k-(k) = N

.

”

(8.61)

C eir*Tir(k) i=l

I STEP 2

3

Fori=l,...

, N, determine the marginal probabilities:

cR

A-(k) ari(j zr

r=l ri(j

1 k)

-

1 I k-

L)eir,

f0r.i

=

l,---,k,

(8.62)

=

for j = 0.

Example 8.7 Consider again the example given in Fig. 8.6 where the multiple-server node 1 is replaced by a general load-dependent server. Loaddependent service rates are: ~~(1) = 1.25, ,u4(1) = 1, ~~(2) = 1.25, ~~(2) = 2, ~~(3) = 1.25, ~~(3) = 3.

,ul(l) = 2, ~~(1) = 1.667, ~$2) = 4, /~(3) = 4,

~~(2) = 1.667, ~~(3) = 1.667,

Initialization:

r;(O ) 0) = 1 for i = 1,2,3,4. Iteration

[I

over the number of jobs in the network starting with

Mean response times, Eq. (8.57): ??i(l) = 1n1(0 Pi(l) Ts(l)

= ~ l

10) = 0.5, n3(0 ( 0) = 0.8,

T2(1) = --&(O TJ(1) =

P3U)

[I

Throughput, X(1) =

4 C i=l

l -7r4(0 Y4P)

I 0) = 0.6, 1 0) = I.

Eq. (8.61) and normalizing constant, Eq. (8.42) : l

eiTi(l)

= 0.454,

G(1) = z

= 2.203.

350

ALGORITHMS FOR PRODUCT-FORM NETWORKS

-1

Marginal probabilities, m(l

Eq. (8.62):

w

1 1) =

7rl(O ( 1) = ~~(0 ( 1) = 7r3(0 ( 1) = n-4(0 1 1) =

------n-1(0 10)el = 0.227, CL1 (1) 1 - 7ri(l ( 1) = 0.773, 0.864, ~~(1 1 1) = 0.136, 0.818, ~~(1 ( 1) = 0.182, 0.545, Q(l ) 1) = 0.454.

Iteration for k = 2: -1

Mean response times: Tr(2) = --7rl(O 1

2 ni(1 I 1) = 0.5, 11) + ___ I-L1 (2)

CL1(1)

2 1)+ -7r2( Tz(2) = 1 P2(1) sTr2(o I-Q(2) 2 T3(2) = J1) + -7r3(1 P3(1) T3(o CL3(2)

1 1 1) = 0.682, ) 1) = 0.946,

2 n4(1 I 1) = 1. T4(2) = J1) + ___ P4(1) T4(o P4 (2)

-1

Throughput X(2) =

4

and normalizing constant: 2

= 0.864,

G(2) = g

= 2.549.

C @i(2) i=l

-1

Marginal probabilities: 742

I 2) =

w

---7ri(l

I 1)er = 0.049,

Plc4

n(l

I 2)

=

m) -7rl(O

1 1)ei = 0.334,

I-Q)

q(O 12) = 1 - 2

7rl(l ( 2) = 0.617.

1=1

n2(2 ) 2) = 0.035, ~~(2 12) = 0.063, n4(2 ( 2) = 0.196, Iteration for k = 3:

7r2(1 12) = 0.224,

x2(0

r3( 1 12) = 0.283, 7r4(l 12) = 0.472,

x3(0 n4(0

12) = 0.741, 12) = 0.654, ( 2) = 0.332.

THE MEAN VALUE ANALYSIS

-1

351

Mean response times: F(3)

= --&do

12) + 2

4

I 2) +

Pl (2) &(3)

= 0.776,

-1

F3(3)

Throughput

-1

TA(3) = 1.

and normalizing

Marginal

1 2) = 0.512,

I-L1 (3)

= 1.127,

X(3) = 1.218,

3 -7r1(2

constant:

G(2) G(3) = -q3)

= 2.093.

probabilities:

v9 = -P1(3)r1(2 w 7r1(2 1 3) = /11(2P Jw I 3) = -rl(O

2)el = 0.015,

7r1(3 13)

v(1

2)el = 0.102, I2)el

= 0.375,

I-L1(1)

3

7rl(O 1 3) = 1 - Cxl(Z

) 3) = 0.507.

1=1

3) = 0.082,

7r2(3

1

0.013,

7rz(2

~(1

] 3) = 0.271,

~~(0

3) = 0.634,

~~(3

)

7r3(2

3) = 0.138,

7r3(1 I 3) = 0.319,

7r3(0

3) = 0.512,

7rITq(313) = 0.080,

7r4(2

3) = 0.287,

7rA(l ] 3) = 0.404,

7rJ(O ] 3) = 0.229.

3) 3)

=

=

0.031,

Now the iteration stops and all other performance measures can be computed. For the mean number of jobs at the nodes we have, for example: x1(3)

13)

0.624,

k=l

-

&,(3) The MVA dence:

=-&q(k = = = 0.473,

is extended

X3(3)

0.686,

- = K4(3)

by [Saue83] to deal with

1.217.

other kinds of load depen-

l

Networks in which the service rate of a node depends on the number of jobs of the different classes at that node (not just on the total number of jobs at the node).

l

Networks

with

load-dependent

routing

probabilities.

352

ALGORITHMS FOR PRODUCT-FORM NETWORKS

In [TuSa85] and [HBAK86], the MVA is extended to the tree mean value anulysis, which is well suited for very large networks with only a few jobs. Such networks arise especially while modeling distributed systems.

8.2.4.2 Mixed Networks The MVALDMX algorithm (mean value analysis load dependent mixed) was first presented in [BBA84] and can be used to analyze mixed BCMP networks with load-dependent service stations. Apart from the algorithm presented in [BBA84], another suggestion is made in [KTK81] to extend the MVA to analyze multiclass load-dependent queueing networks. In order to extend the MVA, the equations derived in the previous sections, and especially the equation for the mean response time, need to be extended. Since a mixed network contains open as well as closed job classes, we need to consider the open classes in the network when computing the performance measures for the closed classes. Furthermore the load-dependency must be explicitly accounted for. To do so we need to consider marginal probabilities in the corresponding equations. Bruell, Balbo, and Afshari [BBA84] introduce the capacity function to describe the behavior of load-dependent service stations. The capacity function ci(j) of node i is the number of jobs completed by that station if there are j jobs in the queue. The inverse function of the capacity function, C(j), dependence factor. The loud-dependent mean service time si,(j) tion i is given by the relation

is called the Zoud-

for class-r jobs at service sta-

(8.63) The function Sir(j) can be interpreted as the nominal service time. Using the capacity function c,(j), we can model a general load-dependency, which means, ci(j) can be any function with: ci : W -+ lR+ and ci(l) = 1. In the following, load-dependency shall be restricted to multiple server nodes. The load-dependency factor for an -/M/m node is then:

1 Ic’ I-

if k < m;,

Ci(k) = 1 = c,(k)

1 , ifkzm;. mi

(8.64)

THE MEAN VALUE ANALYSIS

353

In the MVA for mixed queueing networks, and especially in the MVALDMX algorithm, the procedure is to consider the performance measures for open and closed classes separately from each other. At first the performance measures of the closed classes are computed and then the performance measures of the open classes are computed. This separation can be achieved in the MVALDMX algorithm by checking the marginal probabilities of different job combinations. In [BBA84] a recursive equation is given for the computation of the conditional probabilities ni(lc, K) such that there are exactly Ic jobs of the closed job classes at node i for a given population K. This equation is only valid for the closed job classes. This recursive equation is the only one necessary for the analysis of mixed, load-dependent queueing networks:

T~(JC,K) = 5

s,,cl +ECi(~)

. &cl(K)

+ri(k

- 1, K - lc&

(8.65)

Here, K is the population vector of jobs in the closed classes, and 1,~ is the vector whose components are all zero except the clth component, which is one. The variable Ic is the overall number of jobs in closed classes in a certain population vector during the iteration, and ECi is the inverse of the effective capacity function. The reduction of the mixed network to a closed network is given by modifying the load-dependence factor Ci (Ic). In a mixed queueing network, the service stations are used by jobs belonging to open as well as closed classes. The availability of open job classes in the network reduces the capacities of the service stations, which means that the closed job classes do not have the full capacity available any more as previously defined in the capacity function. It is for this reason that we need to define the eflective capacity function: eciw

1 = Eci(q

= c&c) *

Ei(k - 1) E;(k)

’

(8.66)

This function describes the capacity that is effectively available for the closed classes if we consider the jobs of open classes. This step makes a separate analysis of the open and closed classes possible. As we can see, Ei(k) (which is a function that helps in computing the effective capacity of station i) is of basic importance for this reduction. The computation of the auxiliary parameters can be done for each service station i independently from the other service stations. Vk>mi-

1

:

Ei(k)= Li = E op=l

1

1 _ Li . Ci(mi) ' Ei(k - l)' Xi,op ' Si,op,

354

ALGORITHMS FOR PRODUCT-FORM NETWORKS

Ei(k) = E@) I

I

&l(k)

+ E&k)

1 1 - LiCi(mi)

-

@3(k)

. ci C,(k)

. Eil(k - l),

k > 0,

= ( k = 0,

mi-2

EiZ(k) = c &,(k,j), j=o mi-2 &3(k)

=

c

‘dk 5 rni - 1

F3(k&

j=o

‘k+j j

. Li +Ci(mi) . Fis(k, j - l),

$$

- Fi3(k - l,O), i

mi-lC,(Z) II1=1ciCrni> ’

IC> 0, j > 0, k > 0, j > 0,

k = 0, j = 0.

Here Li denotes the load factor of open classes with regard to station i. Now we have all necessary equations to describe the MVA algorithm for mixed product-form queueing networks with multiple server nodes. In the algorithm, K denotes the overall number of jobs in the closed job classes of the network and k denotes the state vector for an iteration step for the closed job classes. Initialization. the following values: [I

For all nodes i = 1,. . . , N, of the network compute

Load factor Li of the open job classes: Li = E

Xi,*p ’ Si,op*

op=l

Let K = E Kcl and compute for all k = 1, . . . ,K+l cl=1 auxiliary functions Ei(k) and ECi (k).

-1

1 STEP 1.3 1 ni(O, 0) = 1.0.

the

THE MEAN VALUE ANALYSIS

355

Construct a closed model that contains only jobs of closed job classes of the original network and solve this closed model. The results that we get from this model are the performance measures for the closed classes of the mixed network. For all vectors k = 0, . . . , K, do:

I

STEP 2 i For all cl = 1, . . . , CL and i = 1, . . . , N, compute the mean response time:

k

F+(k)

=

““’

cj. ’ j=l

ECi(j)

. ni(j - 1, k - lcl),

Type-3.

Si,cl,

I

STEP 2 2 put:

Type-1,2,4,

For all cl = 1,. . . , CL and i = 1,. . . , N, compute the through-

J&&)

z

N

ICC1’ ei7cz

.

C ej,cz . Tj,dk) j=l

STEP 2 3 jobs:

I

For all cl = 1, . . . , CL and i = 1, . . . , N, compute the number of

(-?EEZGL]

Adapt the marginal probabilities

for Type-1,2,4 nodes:

For i = 1, . . . , N do: sum = 0.0 for k = 1, . . . , k do: n&k)

= E

sZ,CZ. ECi(k) * h,cl(k)

* ri(k - 1, k - lcl),

cl=1

sum = sum + ni(lc, k),

ri(O, k) = 1.0 - sum. 1-1 Compute all other performance measures using the well-known formulae (see Section 7.1) Compute from the solution of the closed model, the performance measures for the open classes of the network. For op = 1, . . . , OP and i =

356

ALGORITHMS FOR PRODUCT-FORM NETWORKS

1,“‘7 N, compute: + 1) . EC@ + 1) . ri(lc, K),

Type-Q,&

Type-3.

k,op * %,op, %,0,(K) Ti,,,(K)

Xi,op

=

’

Type-1’2’47

In the following example, we show how to apply the method to a simple queueing network: Consider the network given in Fig. 8.9. The mean service Example 8.8 times are given as follows: 1 = msec, Pll 1 = @set, P21

-

1

= usec,

-

1

= Qsec,

1

P22

=

usec,

1 -

1

= osec,

Pl4

Pl3

Pl2

-

usec,

=

CL23

1 -

P24

=

@Jsec.

The arrival rates for the open class jobs are Xi = 0.5 jobs/set and X2 = 0.25 jobs/ sec. In each of the closed classes there is K3 = K4 = 1 job. The routing probabilities are: 0.5,

P21,ll

=

1,

= 0.6,

p22,12

=

1,

pl3,23

= 0.5,

pl4,24

= 0.4,

p23,13= 1, p24,14= 1.

PO,11 =

1,

Pll,ll

= 0,

P11,21

=

PO,12 =

1,

P12,12

= 0,

m2,22

PO,13 = 0,

P13,13

= 0.5,

PO,14 = 0,

p14,14

= 0.6,

For both nodes of the mixed network compute:

~1,~~ = X11 . sll

+

X12 - ~12 = 0.9 jobs,

op=1 L2

=

5 op=l

x2,op

-

~2,~~= Xzl + ~21 + X22 - ~22 = Q jobs.

CL

-1

Set: K = C Kc1 = K3 + K4 = 2. cl=1

THE MEAN VALUE ANALYSIS

Fork=

l,...

i=l:

, KcL + 1 compute Ei(k) and ECi(k):

&l(l)

=

'%1(z)

= h(2)+

k(3)

=

ECl(k)

&l(l)

+

&2(l)

+

h(3)

-

&3(l)

=

100,

&z(2) - &s(2)

= 1000,

-

= 10000,

~%2(3)

E13(3)

1 =-=lo, 1 - Ly

= 1 _ L$(ml) 1

i=2:

357

* G(m)

computed in a similar manner as for i = 1.

~~(O,(O,O))

= 7r2(0,(0,0))

= 1.0.

Analysis of the closed model, which we get after removing all open job classes in the network. Iteration for k = (1,O): [STEP]

Compute the mean response times:

‘T13(1,0)

=

. &

s13

.ECl(T)

* @,

(O,O))

r=l

T&1,0) -1

= . . . = 5.

Compute the throughputs: X13(1,0)

=

2

C

k3 *e13

-

ej3

= 0.182,

- Tj3(W

j=l

X23(1,0) = . . . = 0.091. 1-1

-1

Compute the mean queue lengths: rir,,(l,O)

= X13(1,0)'T13(l,0)

K23(l,O)

= ... =

0.455.

Adapt the marginal probabilities: k = “c” kcl = 1. cl=1

= 0.545,

=

3,

358

ALGORlTHMS FOR PRODUCT-FORM NETWORKS

i=l:

sum = 0.0, for Ic = 1: m(L(LO)> = ...

= 5 sl,cl - EG(l) cl=1

- ~l,cz(LO)

.m(O,k

= 0.546,

sum = sum+ rr(l,

(1,O)) = 0.546,

7rl(O, (1,O)) = 1.0 - sum = 0.454,

i=2:

sum = 0.0, for k = 1: 7rz(l, (1,O)) = *. . = 0.454, ~(0, (1,0)) = . . . = 0.546.

Iteration for k = (0,l): -1

Mean response times: T14(0,1) = . . . = 5,

-2.2)

T24(0,1) = . . . = 8.

Throughputs: X14(0,1) = 0.122,

X24(0,1) = 0.049.

-1

Mean queue lengths: K14(0,1) = 0.610, X24(0,1) = 0.392.

11

Adapt the marginal probabilities:

~~(1, (0,l)) 7ry,(1, (0,l))

= 0.610, = 0.390,

7r1(0, (0,l)) x2(0, (0,l))

= 0.390, = 0.610.

Iteration for k = (1,l): L-2.1

Mean response times: T13(1, 1) = 4.829,

T&l,

1) = 6.951,

T14(1, 1) = 7.727, Tz4(1, 1) = 11.636.

- lcl))

THE MEAN VALUE ANALYSIS

-1

359

Throughputs:

-1

&(I,

1) = 0.120,

X&,

1) = 0.081,

X&,1) = 0.060, X2& 1) = 0.032.

Mean queue lengths:

m

&(l,

1) = 0.582,

&(l,

&(l,

1) = 0.624,

K2*(1, 1) = 0.376.

1) = 0.418,

Adapt the marginal probabilities: k = “c” ICC1 = 2. cl=1

i=l:

sum = 0.0, ~~(1, (1,l)) = 0.324, sum = sum + 0.324 = 0.324. 7rr(2, (1,l)) = 0.441, sum = sum + 0.441 = 0.765, rl(O, (1,1)) = 1 - sum = 0.235,

i=2:

sum = 0.0, ~~(1, (1,1)) = 0.332, 7r2(2, (1,1)) = 0.234, 7r2(0, (1,1)) = 0.434.

Now we can compute the performance measures for the open job classes. First the mean number of jobs by class and node:

Kll(1,

K&l)

1)

=

x11

* 311

.~(k+l).~c~(lc+l).rjl(ii,(l,l)) lc=o

= 8.822, = 5.383,

7C12(l,l) = 11.028,

&(l,

1) = 10.766,

Then the mean response time by class and node: T&l)

= “‘;!;)

Tr2(1, 1) = 17.648,

l) = 8.822, T,, (1,1) = 16.766,

T&l,

1) = 28.710.

360

ALGORITHMS FOR PRODUCT-FORM NETWORKS

8.3

THE RECAL METHOD

Another technique for the exact solution of closed product-form queueing networks is the RECAL (recursion by chain algorithm) algorithm. RECAL is very well suited for networks with a large number of job classes but a small number of nodes. For the explanation of this method we assume that we have a product-form network N consisting only of single server or infinite server nodes. We consider only the case of a multiple class network (R > 1) and introduce an extended normalizing constant GR(v) for the network with the Here, GR (0) is the standard normalizing constant vector v = (wi,...,~~). G(K), called G for short in the following. In [CoGe86] a recursive expression for computing G = GR (0) is ’ given, which relates the normalizing constant of a network with r classes and the normalizing constants of a set of networks with each of the (r - 1) classes. The RECAL algorithm uses a simplified version of this general relation. This simplification can be given when each job class contains exactly one job, because if K, = 1 for r = 1,. . . , R, then the normalizing constant GR(O) is recursively given by the expression [CoGe86]: Gr(vr)

= c(1

EG,-l

fvd)

(vr + li)

for vT E F,,

(8.67)

i=l

with:

vr = (‘ulr,...,~Nr), Vir

>Ofori=

, OLr
l,...

r = R,

6i =

1, Type-LW, 0, Type-3,

li = (0,. ss,O,l,O,* v~70)

is an N-dimensional vector with 1 at the ith position.

With the help of the normalizing constant GR (0)) the throughput and the mean number of class-R jobs can be computed because for KR = 1 the correctness of the following equations are proven in [CoGe86]: N

GR-I($) eiR* c j=l

GR(O)*

XiR = e R GR-l(lz) i * GR(“)

(N

+ K

-

1) ’

if there are no Type-3 nodes in the network, for a Type-3 node with index 5,

(8.68)

THE RECAL METHOD

361

and: K,

=

eiR

ih--l(b)

ar

(8.69)

-*%(o)

/-hR’

The basic idea of the RECAL algorithm is to generate a fictitious network containing one job per job class in order to be able to compute the performance measures using Eqs. (8.67), (8.68), and (8.69). Therefore, it must be possible to partition each class T into K, identical subclasses with exactly one job per job class. This partitioning changes the state space and the normalizing constant of the network, but the performance measures remain the same. Let N* denote the fictitious network that we get after partitioning the job classes into subclasses. The overall number of jobs in this fictitious network:

K*=eK,=K, r=l

and the number of job classes is, by definition of N*, exactly R* = K*. Each class r = l,... , R* contains exactly K: = 1 job. Let the jobs be numbered 1,2,. . . , K* and let the function c(k) denote the class index of the lath node in the original network. Using Eq. (8.67), the normalizing constant Gk, (0) of the fictitious network JV is recursively given by:

G;(vk) = $(l

+ vikSi)*~, (k)GLlbk

for Ic= l,...

(8.70)

+ li>T

ZC

i=l

, E(* and vk E .?$ with: N vi,&

2

0

for

i =

1, . . . , N; c

Vik = K* - k

,

i=l

and the initial conditions: G;T(vo) = 1 for all va E .7--i. With Eqs. (8.68) and (8.69) the performance measures A& and ??& relating to a class-c(R*) job in the original network N can be computed: N Gk*-l(1.d eic(R*)’

c j=l G;, (0). (N + K* - 1) ’ f;;~;;;lte,,n~e~w~;

(8.71) \

G&,(L) e4R*Y G;,(o)

’

for a Type-3 node with index 2,

362

ALGORiTHMS FOR PRODUCT-FORM NETWORKS

E&* = G*K*&i) .-eicp) Gil* (0) Pic(R*)*

(8.72)

From the fact that all the jobs of a given class in the network N are identical, the throughputs and the mean number of jobs in class c(R*) of the network N are given by: &c(R’) = Kc(R*)’ &* KiccR*) = Kc(R*&Re.

(8.73)

7

(8.74)

Performance measures for classes other than c(R*) can be obtained by relabeling the jobs in the fictitious network p with the goal that the assignment c(R*) belongs to another class in the original network N. At the beginning, the numbering of jobs in W is chosen with the following assignment to the jobs in N:

1, I

k= l,...,Kr - 1, 2 5 i 5 R and

i,

i-l

k = 1 + c(K7-

- l), . . . , -&G-

I

R

R

R-k+l+C(Kr-l),

- 9,

r=l

r=l

k=l+)(Kr-I),**-yK** r=l

r=l

(8.75)

After computing G&, (0) using Eq. (8.70), performance measures for class s = 1 in the original network N can be determined using Eqs. (8.71)-(8.74), due to Eq. (8.75). For the computation of performance measures of class(s + 1) jobs in the network N, the jobs in N* have to be relabeled afterwards in the following way: R .,S-1+x

(Kr

-

I>,

r=l e(Kr r=l

-

l+e(Kr-

(8.76)

I>,

l),...,K*.

r=l

Now, the performance measures for class-(s + 1) jobs can be computed after computing G&, (0) anew. The RECAL algorithm can be described in the following four steps:

THE RECAL METHOD

363

Initialization: Gc(ve) = 1 for all vo E .7$. Number the jobs in the fictitious assignment c(l)(k) given in Eq. (8.75).

network n/* according to the

Compute and store the values of Gz(v,) - R by using Eq. (8.70).

for all w, E .7$ with

Iteration over all classes s = 1,. . . , R in the network M -1 Compute Gk, (0) with Eq. (8.70) by using the stored values for the Gi (v~). -1 Compute the performance measures of class s using Eqs. (8.71)(8.74). If s = R then the algorithm stops. 1-1 For computing the performance measures of class-s + 1 jobs, relabel the jobs in P corresponding to the assignment (8.76). 1-1 Increase x by 1 and compute and store the values of the Gc(v,) anew for all v, E -Tjc*by using Eq. (8.70). By storing the values Gz(v5) computed in Step 3 and Step 4.4, it is not necessary to compute Eq. (8.70) in Step 4.1 anew for all k = 1,. . . , K* when evaluating the performance measures. Instead, Eq. (8.70) needs to be calculated only for k = x, . . . , K* with x = s + C,“=, (K, - 1). The algorithm is illustrated by the following example: Consider a closed queueing network with N = 2 nodes and Example 8.9 R = 3 job classes and with K = 5 jobs distributed as follows: K1 = 2, KS = 1, KS = 2. We have K = R* = K* = 5. Class switching is not allowed. Node 1 is a Type-2 node and node 2 is a Type-l node. The service rates are given as follows: Pll

=

20%

p12

=

50,

p13

= 2,

p21 = p22 = p23 1 4,

and the visit ratios are: ell = e12 = e13 = 1,

e21 = 0.72,

e22 = 0.63,

e23 = 0.4.

The computation of the performance measures is carried using the RECAL algorithm as follows: Initialization: Gg(ve) = 1 for all ve E Fi, with F; =

364

ALGORITHMS FOR PRODUCT-FORM NETWORKS

Labeling the jobs in the network w k k k k k

= = = = =

with Eq. (8.75) yields: 1, 2, 3, 4, 5.

Compute Gz(w,) for all v, E ,FG and x = K* - R = 2. We get:

zli2>0,

= uo, % (19% (2, I>, (3,0>> *

&=K*-2=3 i=l

With Eq. (8.70) we have: $

G;(l, 3) + 4.2.

G;(O, 4) = 0.727,

G;(1,2)

= 2.2.

G;(2,2) + 3.2.

G;(l, 3) = 0.774,

G;(2,1)

= 3.2.

G;(3,1) + 2.:.

G;(2,2)

G;(3,0)

= 4$.G;(4,0)

G;(O, 3) =

+

E.G;(3,1)

= 0.681, = 0.448,

0.905,

z.

G;r(l, 4) + 5.2.

G;(O, 5) =

G;(l, 3) = 2.2.

G;(2,3) + 4.2.

G;(l, 4) = 0.73,

G;(2,2)

= 3.2.

G;;(3,2) + 3.2.

G;;(2,3) = 0.555,

G;(3,1)

= 4.E.

G;(4,1) + 2.2.

G;(3,2)

= 0.38,

G;(4,0)

= 5.2.

G;(5,0) +

2.

G;(5,0)

= 0.205.

GT(O, 4) =

Iteration with s = 1:

over all job classes s of the given network JV, starting

THE RECAL METHOD

of G:(O) using Eq. (8.70). For this computation

I] Computation we additionally need: G;(O, 2) =

E-G;(l,2)

G;(l, 1) = 2$G;(2,1) = 3-E.

G;(2,0)

G;(O, 1) = G:(l,O)

365

E.

+3 ~2.

G;(O, 3) = 0.605,

+ 2.2. G;(l, 2) = 0.836, CL23 e23. G;(2,1) = 0.740, /-k23

G;(3,0) +

G;(l, 1) + 2-E.

= 2.$.Gj(2,0)

G;(O, 2) = 0.207,

“22-G;& Pm

+

1) = 0,161.

From this i t follows: G;(O) = E,

G;(l, 0) + 2.

G;(O, 1) = 0.038.

1-1 Computation of the class 1 performance measures. Because of Eqs. (8.71) and (8.72) we have: 2

XT5= ellsC j=l

GXlj) + 5 - 1) = 1.6117

G;(o)-(2

2

AZ5= e2,+C j=l

Wlj)

G;(o). (2 + 5 - 1) = m7 2

= 0.021,

G;(O) 1). 2 -* K25 = -- 1 G;(O)

= 0.979.

G;(l,O)1 -* I-Cl5 = -.G;(O)

These results are needed to compute the throughputs jobs in class 1 using Eqs. (8.73) and (8.74). = 3.222,

xl1 = &.l;i;,

= 0.042,

(8.77)

X21 = K1. A;, = 2.320,

1721 = Q??;,

= 1.958.

(8.78)

Xl1 = K14;5

-1

and mean number of

Relabel the jobs in JV* in accordance to Eq. (8.76): 17

37

d2)(k) =

17

I

3, 27

k = 1) k = 2) k = 3) k =4, k = 5.

366

ALGORITHMS FOR PRODUCT-FORM NETWORKS

[ml Increase x by 1, i.e., x = 3 and compute and store Gz (vz) for all v, E Fz = {(0,2), (1, l), (2,O)):

G;(0,2) =

$G;(l,2)

+3=G;(O,3)

= 0.396,

k421

G;(l, 1) = 2.2.

G;(2,1) + 2.2.

G;(l, 2) = 0.285,

Pll

G$(2,0) = 3$.G;(3,0)+

G;(2,1) = 0.129.

$

Iteration for s = 2: I-$,11 additionally

Computation of G;(O) with Eq. (8.70). For this computation we need: E.Gj(l,

1) + 2 .z.G;(0,2)

= 2$.G;(2,0)+

2.

G;(O, 1) = G;(l,o)

= 0.222,

G;(l, 1) = 0.158.

Hence: G;(O) =

z.G;(l,O)

im Computation Eq. (8.71) and Eq. (8.72):

+

E.G;(O,

1) = 0.038.

of the performance measures for class 2 using

A& = e12-C2 ~Gm jzl Gw

6

= 1.661,

Furthermore: A;, = 1.046,

K;,

= 0.083,

E& = 0.917.

Because of K2 = 1 with Eqs. (8.73), (8.74) we get: x12 = A;,,

mq

x22 = A;,,

E,,

= IT&,

Relabel the jobs: 1, fk = 1,

3, cqk)

k =

2,

= I 1, k = 3,

2, 3,

k = k =

4, 5.

7722 =

I?;,.

THE RECAL METHOD

11 Increase z by 1, i.e., x = 4, and compute and store Gz(v,) all v, E -T4*= ((0, l), (1,O)): z+

G:(O, 1) =

G;(l, 1) + 2.2.

‘%(I, 0) = 2$.G;(2,0)

+

367

for

G;(O, 2) = 0.131,

z.G;(l,

1) = 0.050.

Iteration for s = 3: -1

Computation

of GE(O):

G;(O) = E.G;(l,O) 1-1

Computation

+ $G;(O,

1) = 0.038.

of the class-3 performance measures:

XT:, = 0.790,

A;, = 0.316,

rr5

= 0.658,

?7& = 0.343.

Xl3 = 1.580,

XZ3 = 0.632,

El3 = 1.315,

K15 = 0.685.

Hence:

Now the iteration ends and all other performance measures can be obtained using formulae from Section 7.1. If the number of job classes in the network is very large, then the RECAL algorithm is much more efficient than the convolution algorithm or the MVA. But if there are only a few job classes and many nodes in the network, then RECAL is not very well suited (see Figs. 8.10 and 8.11). If we assume that K, = K for all T and N is fixed, then the time complexity and memory requirement grow polynomially with the number of job classes in the network. In this case, the memory requirement is:

and the time requirement in number of operations is:

0( (y-$RN+’ >. The algorithm can be extended further to analyze Type-l nodes with multiple servers and to analyze nodes with load-dependent service rates. Based on the RECAL algorithm, [CSL89] developed the MVAC algorithm (mean value analysis by chain), which directly computes the mean values of performance measures. The RECAL method was extended to the tree RECAL technique by [McKe88].

368

ALGORITHMS FOR PRODUCT-FORM NETWORKS classes (R)

18 16 14 12 10

8

(NJ fig. 8.10 Regions in the (N x R) space in which the storage requirement of RECAL is less than that of convolution, for K = 1 and 12 (on or above the curves the storage requirement of RECAL is less than that of convolution).

8.4

FLOW EQUIVALENT

SERVER METHOD

The last method for analyzing product-form queueing networks that we consider is called flow equivalent server (FES) method. The method is based on Norton’s theorem from electric circuit theory [CHW75b]. We encounter this method again in Chapter 9 when we deal with approximate solution of non-product-form networks. Here we consider this technique in the context of exact analysis of product-form networks. If we select one or more nodes in the given network and combine all the other nodes into an FES, then Norton’s theorem says that the reduced system has the same behavior as the original network. 8.4.1

FES Method for a Single Node

To describe the FES method, we consider the central-server model given in Fig. 8.12 where we suppose that a product-form solution exists. For this network we can construct an equivalent network by choosing one node (e.g., node 1) and combining all other nodes to one FES node c. As a result we get a reduced net consisting of two nodes only. This network (shown in Fig. 8.13) is much easier to analyze then the original one, and all computed performance measures are the same as in the original network. To determine the service rates ~Jlc), k = 1,. . . , K of the FES node, the chosen node 1 in the original network is short circuited, i.e., the mean service time of this node is set to zero, as shown in Fig. 8.14. The throughput in the short circuit path with k = 1,. . . , K jobs in the network is then equated to

FLOW EQUIVALENT SERVER METHOD

369

classes (R) v/c=1

22

RECAL

20 f

I’ ,,

/’

18 16 14 12 10 8 6 4 2 2

4

6

8

10

12

Fig. 8.11 Regions in the (N x R) space in which the time requirement of RECAL is less than that of convolution, for K. = 1 and 12 (on or above the curves the time requirement of RECAL is less than that of convolution).

Fig. 8.12

The central-server network.

the load-dependent service rate ,u~(Ic) , Ic = 1, . . . , K of the FES node in the reduced queueing network. The FES algorithm can now be summarized in the following three steps: In the given network, choose a node i and short-circuit it by setting the mean service time in that node to zero. Compute the throughputs XT(k) along the short circuit, as a function of the number of jobs k = 1, . . . , K in the network. For this computation, any of the earlier solution algorithms for product-form queueing networks can be used. From the given network, construct an equivalent reduced network only of the chosen node i and the FES node c. The visit ratios in both nodes are ei. The load-dependent service rate of the FES node is the throughput along the short-circuit path when there are Ic jobs in the network, that is: p,(k) = X?(k) for F; = 1,. . . , K. Compute the performance measures in the reduced network with networks (e.g., convolution or MVA).

any suitable algorithm for product-form

370

ALGORITHMS FOR PRODUCT-FORM NETWORKS

chosen

node FES node

fig. 8.13

Reduced queueing network.

Fig. 8.14

The short-circuited

model.

The technique just described is very well suited for examining the influence of changes in the parameters of a single node while the rest of the system parameters are kept fixed. To clarify the usage of the FES method, we give the following simple example:

Fig. 8.15

Yet another closed queueing network.

Example 8.10 Consider the closed single class product-form network shown in Fig. 8.15. It consists of N = 4 nodes and K = 2 jobs. The service times are exponentially distributed with parameters: fh=l,

The routing probabilities are

p2 =

2,

given

as:

pug=

3, puqz4.

0.5, $I42 = 0.4, = 0.5, p43 = 0.4,

P12 = 0.5,

P2l = O-5,

P31 =

P13 = 0.5,

p24 = 0.5,

p34

p44 = 0.2.

FLOW EQUIVALENT SERVER METHOD

371

With Eq. (7.5) we compute the visit ratios: el = 1,

e2 = 1,

e3 = 1,

e4 = 1.25.

Now, the analysis of the network is carried out in the following three steps: Choose a node, e.g., node 4 of the network shown in Fig. 8.16, and short circuit it. The throughput of the short circuit is computed for Ic = 1,2

Fig. 8.16

Network after node 4 is short circuit.

by using the MVA, for example. Then we get the following results: X(1) = 0.545,

X(2) = 0.776.

XT(l) = 0.682,

x7(2) = 0.971.

and therefore:

Construct the reduced network that consists of the selected node 4 and the FES node c as shown in Fig. 8.17. For the service rates of the FES

Fig. 8.17

Reduced network.

node we have: ,uc(l) = XT(l) = 0.682,

~~(2) = X7(2) = 0.971.

The visit ratios are: e4 = e, = 1.25.

ALGORITHMS FOR PRODUCT-FORM NETWORKS

372

Analyze the reduced network shown in Fig. 8.17 using, for example, the MVA for load-dependent nodes. The load-dependent service rates of node 4 can be computed by Eq. (7.49) as pq (1) = ,Q (2) = 4. MVA-Iteration

for Ic = 1:

T&) = -7rTTq(O 1

( 0) = 0.25,

T,(l)

=

P40)

10) = 1.467,

l-4)

X(1) = 0.466, ~~(1 1 1) = 0.146, MVA-Iteration

1 -n,(O

G( 1) = - ’ = 2.146, w n,(lll) = 0.854, 7r4(0 11) = 0.854,

7r,(OIl) = 0.146.

for Ic = 2: T&)

= -

1 P4W

x4(01

1>+

2 -7r4(l

( 1) = 0.286,

P4 (2)

2 ;?;c(a>= - 1 -ire(W) = 1.974, + -7r,(l(l) PC(1) PC(2) G(l) = 3.032, X(2) = 0.708, G(2) = x(a> ~~(112) = 0.189,

n4(212) = 0.032,

7@12) = 0.189,

7rc(2[2) = 0.779,

~(012) = 0.779, 51r,(O12)= 0.032.

Now all other performance measures of the network can be computed: l

Mean number of jobs: For node 4 we get from Eq. (7.26): IT;4 = 2

Jc’7r4(Ic12) = 0.253.

k=l

For the other nodes we have by Eq. (8.16):

l

772 = 0.436,

x3 = 0.273.

X3 = 0.708,

X4 = 0.885.

Throughputs, Eq. (8.14): = 0.708,

X2 = 0.708,

a Mean response times, Eq. (7.43):

T1= 5 = A? 1466 Xl

T2 =

0.617,

T, = 0.385.

FLOW EQUIVALENT SERVER METHOD

8.4.2

FES Method

for Multiple

373

Nodes

The usage of the FES method, when only one node is short-circuited, has nearly no advantage in reducing the computation time. Therefore [ABP85] suggest an extension of the concept of [CHW75b]. For this extension, the closed product-form network is partitioned in a number of subnetworks. Each of these subnetworks is analyzed independently from the others, i.e., the whole network is analyzed by short-circuiting the nodes that do not belong to the subnetwork that is to be examined. The computed throughputs of the shortcircuited network form the load-dependent service rates of FES node j, which represents the jth subnetwork in the reduced network. When analyzing this reduced network, we get the normalizing constant of the whole network. The normalizing constant can then be used to compute the performance measures of interest. The FES method for this case can be described in the following five steps. -1 Partition the original network into &! disjoint subnetworks SN-j (1 5 j 5 &!). It is even allowed to combine nodes in the subnetwork that do not have a direct connection to this subnetwork. -4 Analyze each of these subnetworks j = 1, . . . , M by short-circuiting all nodes that do not belong to the considered subnetwork. The visit ratios of the nodes in the subnetworks are taken from the original network. For analyzing the subnetworks, any product-form algorithm can be used. Determine the throughputs &iv-j (k) and the normalizing constants Gj (Ic), for Ic = 1, . . . , K. -C om b ine the nodes of each subnetwork into one FES node and construct an equivalent reduced network out of the original network by putting the FES nodes (with visit ratio 1) together in a tandem connection. The loaddependent service rates of the FES node j are identical with the throughputs in the corresponding jth subnetwork: I.Lcj(k) =

for j = 1,. . . ,&! and Ic = I,. . . ,E(.

b~-j(k),

r@Jm] The normalizing constant of the reduced network can be computed by convoluting the 111normalizing vectors

of the subnetworks [ABP85]: G=G1@G2cc..@GM,

(8.79)

where the convolution operator @ is defined as in Eq. (8.1). To determine the normalizing constant, the MVA for networks with load-dependent nodes (see Section 8.2.4) can be used instead.

374

ALGORITHMS FOR PRODUCT-FORM NETWORKS

[m] Compute the performance measures of the original network and the performance measures of the subnetworks with the help of the normalizing constants and the formulae given in Section 8.1. The FES method is very well suited if the system behavior for different input parameters needs to be examined, because in this case only one new normalizing constant for the corresponding subnetwork needs to be recomputed. The normalizing constants of the other subnetworks remain the same. Example again.

The queueing network from Example 8.10 is now analyzed

8.11

ar tti ion the whole network into M = 2 subnetworks where SubIW-JP network 1 contains nodes 1 and 2 and Subnetwork 2, nodes 3 and 4 of the original network.

Fig. 8.18

Subnetwork 1 with subnetwork 2 short-circuited.

Imj Analyze the two subnetworks: To analyze Subnetwork 1, nodes 3 and 4 are short-circuited (see Fig. 8.18). We use the MVA to compute for Ic = 1,2 the load-dependent throughputs and normalizing constants of this subnetwork: XsN-1(1) = 0.667, G1(l) = 1.5,

XSN-r(2) = 0.857, Gr(2) = L.75.

To analyze Subnetwork 2, nodes 1 and 2 are short-circuited

(see Fig. 8.19).

Analogously, we get the following results for Subnetwork 2: XsN-&)

MI

= 1.548,

XSNM2(2) = 2.064,

G2(1) = 0.646,

G2(2) = 0.313.

Construct the reduced network shown in Fig. 8.20.

FLOW EQUIVALENT SERVER METHOD

fig. 8.19

Short-circuit

Fig. 8.20

375

of Subnetwork 1.

Reduced network.

The first FES node describes the nodes of Subnetwork 1 with service rates /car = XSN-r(Ic) for Ic = 1,2, and the second FES node describes the nodes of Subnetwork 2 with service rates p,I;z(k) = XSN-z(JG). The normalization using Eq. (8.79):

constants of the reduced network are computed

By using Eq. (8.1) we get: G(l) = G(O)G(l)

+ Gr(l).Gz(0)

= 2.146,

G(2) =

+ G(l)G(l)

+ Gr(2).G2(0)

WWa(2)

= 3.032.

Compute the performance measures: Mean number of jobs at each node is computed using, Eq. (8.16): = 1.038,

E2 = 0.436,

r3 = 0.273,

hr4 = 0.253. As we can see, the same results are obtained as in Example 8.10. The FES method can also be applied to the case of multiple class queueing networks. The method is very flexible in the need for resources and lies between the two extremes: MVA and convolution. This flexibility is based on how the subnetworks are chosen.

376

ALGORITHMS FOR PRODUCT-FORM NETWORKS

Table 8.5 Storage requirement in number of storage elements and time requirement in number of operations [CoGe86] Method

Storage Requirement

Convolution

Time Requirement O(2. R(N - 1). fi (K, + 1))

O( fi WY + 1)) r=l

r=l

MVA

= nN+l + 1 o$-=-ijT-

RECAL* FES

> O(3 * &K,

O(2.

R(N - 1). &G

-=-L(R(N O( (N + l)!

+ 1)) + 1))

i=2O(2 * R(N - 1). fi (K, + 1)) ?-=l

+ 1))

* Kr=KforT=l,...,R

8.5

SUMMARY

As shown in Table 8.5, the time requirement for convolution, MVA and FES is approximately the same and differs for RECAL in dependency of the number of nodes and number of classes (see also Fig. 8.11). Also, the storage requirement depends greatly on the number of nodes and classes for RECAL compared to the other methods (see Fig. 8.10). For larger numbers of nodes, MVA is much worse than FES and convolution. For FES, the storage requirement depends also on the number of subnetworks. The larger the number of subnetworks, the smaller is the storage requirement. In Table 8.6 the main advantages and disadvantages of the convolution algorithm, the MVA, the RECAL met hod, and the FES method are summarized. For the closed queueing network of Problems 7.3 and 7.4, Problem 8.1 compute the performance measures using the convolution method. Problem 8.2 For a closed queueing network with N = 2 nodes and R = 2 job classes, compute the performance measures for each class and node using the convolution method. Node 1 is of Type-3 and node 2 is of Type-2. Each class contains two jobs Ki = K2 = 2 and class switching is not allowed. The service rates are given as follows:

PII = 0.4sec-l,

p21 = 0.3seCl,

~12 = 0.2sec-‘,

p22 = 0.4sec-1,

The routing probabilities are: Pll,ll

= p11,21 =

0.5,

P21,ll

= p22,12 = p12,z

=

1.

Problem 8.3 Compute performance measures of the network in Problems 7.3 and 7.4 again, with the MVA.

SUMMARY

Tab/e 8.6

Comparison

of the four solution

Method

methods

for product-form

377

networks

Advantages

Disadvantages

MVA

Mean values can be computed directly without computing the normalizing constant. But if required, the normalization constant and hence state probabilities can also be computed.

High storage requirement (for multiple classes and multiple servers) Overflow and underflow problems when computing the marginal probabilities for -/M/m nodes

Convolution

Less storage requirement MVA

than

Normalizing constant (NC) has to be computed Overflow or underflow problems when computing the NC

RECAL

Less storage and time requirement for a large number of job classes and nodes than MVA and convolution

More storage and time requirement for small number of job classes and nodes than MVA and convolution

FES

The time and storage requirement is reduced considerably when examining the influence of changing the parameters of an individual or a few nodes while the rest of the system parameters remain unchanged Basis for solution techniques for non-product-form networks

Multiple application of convolution or MVA Network needs to be transformed

Problem 8.4 For a mixed queueing network with N = 2 Type-2 nodes and R = 4 classes (see Fig. 8.21), compute all performance measures per node and class. The system parameters are given as follows: X1 = 0.2 see-‘,

X2 = 0.1 set-‘,

pll = 1.5sec-l,

~12 = lsec-l,

~21 = 2 set-‘,

p22

P21,ll

= P22,12

PO,11=

PO,12

= P21,o

=

Pll,Zl

=

~13 =

= 0.5 set-l,

= P22,o = Pl2,22

=

K2 = KS = 2,

/L23

3sec-l,

= 2 set-l,

~14 = /L24

2sec-l,

= 1 set-l,

0.5. P13,23

=

P14,24

=

P23,13

=

P24,14

= 1.

Problem 8.5 Consider a closed queueing network, consisting of a Type-3 node and a Type-l node (m2 = 2). This model represents a simple twoprocessor system with terminals. The system parameters are given as: 1 1 K = 3, ~12 =pal = 1, - = 2sec, - = lsec. p2 Pl. (a) Apply the mean value analysis to determine the performance measures of each node, especially throughput and mean response time of the terminals.

378

ALGORITHMS FOR PRODUCT-FORM NETWORKS

Source cl ‘- t --w I I L

-

q&-@-~--, I

i cl Sink

Fig. 8.21

Mixed queueing network.

Q4 Determine

for both nodes the load-dependent service rates and analyze the network again, using the load-dependent MVA.

Problem 8.6 Compute the performance measures for the network given in Problem 8.2 using the RECAL algorithm. Problem 8.7 Consider a simple central-server model with N = 3 Type-l nodes (mi = 1) and K = 3 jobs. The service rates are given as follows: pl = lsec-l,

p2 = 0.65 set-‘, p3 = 0.75 set-‘,

and the routing probabilities are: ~12 = 0.6,

~21 = ~31 =

1,

p13 =

0.4.

Combine nodes 2 and 3 into a FES node. Use the FES method and compare the results with the results obtained without the use of FES method. Consider a closed queueing network with N = 4 Type-l Problem 8.8 nodes and K = 3 jobs. The service rates are: p1 =

4sec-l ? p2 = 3sec-l,

p3 =

2sec-l

7

CL4

= lsec-l,

and the routing probabilities are: p23 =p24

= 0.5,

P12 = P31 = p41 =

1.

For computing the performance measures, use the extended FES method by combining nodes 1 and 2 in subsystem 1, and nodes 3 and 4 into subsystem 2.

9 Approximation Algorithms for Product-Form Networks In Chapter 8, several efficient algorithms for the exact solution of queueing networks are introduced. However, the memory requirements and computation time of these algorithms grows exponentially with the number of job classes in the system. For computationally difficult problems of networks with a large number of job classes, we resort to approximation methods. In Sections 9.1, 9.2, and 9.3 we introduce methods for obtaining such approximate results. The first group of methods is based on the MVA. The approximate methods that we present need much less memory and computation time than the exact MVA and yet give very accurate results. The second approach is based on the idea that the mean number of jobs at a network node can be computed approximately if the utilization pi of this node is known. The sum of all the equations for the single nodes leads to the so-called system equation. By solving this equation we get approximate performance measures for the whole queueing network. In some cases it is enough to have upper and lower bounds to make some statements about the network. Therefore, the third approach (Section 9.4) deals with methods on how to get upper and lower bounds on performance measures of a queueing network. In Section 9.5 we introduce a method that allows intervals as input parameters. For other techniques that have been developed to analyze very large networks see [ChYu83, SLM86, HsLa89].

379

380

9.1

APPROXIMATION ALGORITHMS FOR PRODUCT-FORM NETWORKS

APPROXIMATIONS

BASED

ON THE

MVA

The fundamental equation of the MVA (8.31) describes the relation between the mean response time of a node when there are k jobs at the node and the mean number of jobs in that node with one job less in the network. Therefore, to solve a queueing network it is necessary to iterate over all jobs in the network, starting from 0 to the maximum number K. For this reason, an implementation of the MVA requires a lot of computation time and main memory. For multiclass queueing networks with a large number of jobs in the system, we will very quickly reach the point at which the system runs out of memory, especially if we have multiserver nodes. An alternative approach is to approximate the fundamental equation of the MVA in a way that the mean response time depends only on K and not on K - 1. Then it is not necessary to iterate over the whole population of jobs in the system but we have to improve only the performance measures iteratively, starting with an initial vector. This approach leads to a substantial reduction in computation time and memory requirement. In the following, two techniques based on this idea, are introduced. 9.1.1

Bard-Schweitzer

Approximation

Bard and Schweitzer [Bard79, Schw79] suggested an approximation of the MVA for single server queueing networks that is based on the following idea: Starting with an initial value Ei,(K) for the mean number of class-r jobs at node i for a given population vector K, make an estimate of the mean number of jobs for a population vector (K - 1,). This estimating needs to be done for all classes s. These Ki,(K - ls) estimates are then used in the MVA to compute the mean response times, and only one iteration step is now needed to evaluate the performance measures for a given population vector K. In particular, new values for the K;,(K) are computed. The values we get are used to estimate the K+(K - 1,) again. This iteration stops if the values of the Ki,.(K) in two iteration steps differ by less than the chosen error criterion E. Obviously, the problem with this MVA approximation is how to estimate the mean value xi, (K - I,), given xi, (K) . An approximate formula for the case of a very large network population was suggested by [Schw79] since in this case a reasonable explanation can be given as to why the mean number of jobs of class T at node i remains nearly the same if the number of jobs in the network is reduced by one. For the mean number of class-r jobs at node i given a population vector (K - ls), they suggest the approximation:

x,r(K

-

1s)

=

cK

- 'd,( K r

zr

(K)

y

(94

APPROXIMATIONS

BASED ON THE MVA

381

where:

P-2)

:fE' ,

'

gives the number of class-r jobs when the network population K is reduced by one class-s job. If we assume that the jobs are initially equally distributed over the whole network, then the Bard-Schweitzer approximation can be described in the following steps: 1-1

For i = 1,. . . , N and s = 1,. . . , R:

Initialization.

r?,,(K)

= $.

[wa For all i = 1,. . . , N and all r, s = 1,. . . , R, compute the estimated values for the mean number of jobs for the population vector (K - 1,): Kis(K

- 1,) = (K G1””

*Eis(K).

S

-\A na 1yze the queueing network for a population vector K by using one iteration step of the MVA.

I STEP 3

1 For i = 1, . . . , N and T = 1, . . . , R, compute the mean response

times:

1 +&i,(K

-

- 1,)

s=l

I1

j&r

I STEP 3

1 ,

Type-l&t

(mi = I),

Type-3. ’

2 Forr=l,...

, R, compute the throughputs: X,(K)

=

N

Kr

.

C ?Tir(K)eir i=l

I STEP 3

3 For i = 1, . . . , N and T = 1, . . . , R, compute the mean number

of jobs:

K,,.(K)

= Ti,.(K)X,.(K)ei,.

I/ Check the stopping condition: If there are no significant changes in the xi, (K) values between the nth and (n - 1)th iteration step, that is, when: max K!:)(K) i,r

- K~~-l)(K)~

< E,

382

APPROXIMATION ALGORITHMS FOR PRODUCT-FORM NETWORKS

for a suitable E (here the superscript cn) denotes the values for the nth iteration step), then the iteration stops and all other performance values are computed. If the stopping criterion is not fulfilled, then return to Step 2. This approximation is very easy to program and faster than the exact MVA. The memory requirements are proportional to the product of N and R. Therefore, this approximation needs considerably less memory than the exact MVA or the convolution method, especially for networks with a large number of job classes. A disadvantage of this approximation is that networks with multiple server nodes cannot be solved using it. We give a simple example of the use of the algorithm: We revisit the single class network of Example 8.2 (Fig. 8.4) Example 9.1 with network parameters as shown in Table 9.1. For & we choose the value 0.06. The analysis of the network (K = 6) is carried out in the following Table 9.1 Input parameters for Example 9.1. i

ei

l/Pi

1 2 3 4

1 0.4 0.2 0.1

0.02 0.2 0.4 0.6

7%

1 1 1 1

steps: Initialization: 17,(K) = K&x-)

= 11;;#iy = T-c@) = g =l&

1. Iteration: Estimated values, Eq. (9.1): Ir,(K

- 1) = ~.7qK)

rjz(K

- 1) = K&c

= 1.25, - 1) = Kq(K

- 1) = 1.25.

One MVA iteration: 1-1 ?;@c) =;

Mean response times: [1+ Ei(K

- 1)] = 0.045,

T2(K) T/&+-$1

= ;

[I + 1T,(K - l)] = 0.45, +i7,(K

- l)]

= 1.35.

APPROXIMATIONS

-4

BASED ON THE MVA

Throughput:

X(K)= 4 K

= 11.111.

C eiTi(K) i=l

I

Mean number of jobs: xl(K)

= Fl(K)X(K)el

= 0.5,

E,(K)

= Tz(K)X(K)ez

= 2,

x,(K)

= Ts(K)X(K)es

= 2,

zd(K)

= FA(K)X(K)eA

= 1.5.

mj

Check the stopping condition: max IKjl)(K) i

- $“(K)l

> 0.06.

= 1

2. Iteration: Estimated values: xl(K

- 1) = y.

0.5 = 0.417,

K2(K - 1) = 1.667, mi?ma

K3(K - 1) = 1.667,

K4(K

- 1) = 1.25.

One MVA Iteration: ?&(K) = ;

[l+

T2(K) = 0.533, X(K) =

4

K1(K - l)] = 0.028 Fs(K)

= 1.067,

T4(K) = 1.35,

= 10.169,

K

C eiTZ(K) i=l

Lmj

XI(K)

= Tl(K)X(K)el

x2(K)

= 2.169,

= 0.288,

Es(K)

= 2.169,

r4(K)

= 1.373.

Stopping condition: max E{“’ (K) - $l’(K) i

1 = 0.212

> 0.06.

3. Iteration: [w]

Estimated values: j&(K E,(K

- 1) III y.

0.288 = 0.240, - 1) = 1.808, K3(K - 1) = 1.808,

c4(K

- 1) = 1.144.

383

APPROXIMATION

384

ALGORITHMS FOR PRODUCT-FORM NETWORKS

MVA Iteration: T&T)

= 0.025,

T2(K)

= 0.562,

T3(K)

= 1.123,

??#i-) = 1.286,

X(K) = 9.955, ??,(A-) = 0.247, wm@

K2(lX-) = 2.236, I,

= 2.236, x4(K)

= 1.281.

Stopping condition: max zd3’(K) i

- $“‘(K)I

= 0.092

> 0.06.

4. Iteration:

li;,(K

- 1) = 0.206,

K2(K

- 1) = 1.864,

K3(K

- 1) = 1.864,

??.*(K - 1) = 1.067.

[y?#p@gj T&T) = 0.024, X(K) = 9.896,

T2(K)

= 0.573,

T3(K)

= 1.145,

T4(K)

= 1.240,

77,(K)

K2(K)

= 2.267,

x3(K)

= 2.267,

E4(K)

= 1.227.

= 0.239,

m max $‘)(K) i

- f?i3’(K)I

= 0.053

< 0.06.

Now, the iteration stops and the other performance measures are computed. For the throughputs of the nodes we use Eq. (8.40): X1 = X(K). el = 9.896, For the utilization

X2 = 3.958,

X3 = 1.979,

X4 = 0.986.

of the nodes we get:

pl = K = 0.198,

p2 = 0.729,

p3 = 0.729,

p4 = 0.594.

The final results are summarized in Table 9.2. A comparison with the values from Example 8.2 shows that the BardSchweitzer approximation yields results that in this case are very close to the exact ones. The storage requirement of the Bard-Schweitzer approximation depends only on the number of nodes N and the number of classes R (= O(N x R)) and is independent of the number of jobs. This is much smaller than those for convolution and MVA.l The accuracy of the Bard-

lRecal1 0

N.

that fi r-=1

the storage

p&+1)

>

requirements

) respectively.

for convolution

and MVA

are 0

fi (K, r=l

and

+ 1) >

APPROXIMATIONS

Tab/e 9.2

BASED ON THE MVA

385

Bard-Schweitzer approximation for Example 9.1 1

2

3

4

0.024 9.896 0.239 0.198

0.573 3.958 2.267 0.729

1.145 1.979 2.267 0.729

1.240 0.986 1.240 0.594

Node: Mean response time T, Throughput A, Mean number of jobs E; Utilization pZ

Schweitzer approximation is not very good (the mean deviation from exact results is approximately 6%) but is sufficient for most applications. An essential assumption for the Bard-Schweitzer method is that the removal of a job from a class influences only the mean number of jobs in that class and not the mean number of jobs in other classes (see Eq. (9.1)). This assumption is not very good and we can construct examples for which the computed results deviate considerably from the exact ones. In Section 9.1.2, the self-correcting approximation technique (SCAT) is introduced, which does not have this disadvantage. Of course, the algorithm is more complex to understand and implement than the Bard-Schweitzer approximation. 9.1.2

Self-Correcting

Approximation

Technique

Based on the basic idea of Bard and Schweitzer, Neuse and Chandy [NeCh81, ChNe82] developed techniques that are an improvement over the Bard-Schweitzer approximation [ZESSS]. Th e main idea behind their approach is demonstrated on queueing networks consisting only of single server nodes. 9.1.2.1 Single Server Nodes The SCAT algorithm for single server nodes [NeCh81] gives better results than the Bard-Schweitzer approximation, especially for networks with small populations, because in the SCAT approximation we not only estimate the mean number of jobs but also the change in the mean number of jobs from one iteration step to the next. This estimate is used in the approximate formula for the Ei, (K - 1,). We define F&. (K) as the contribution of class-r jobs at the ith node for a given population vector K as: &r(K)

=

E,,(K) K. r

If a class-s job is removed from the network, Dir,(K) contribution: Dirs(K)

= Fir(K =

Fir(K

gives the change in this

- 1,) - Fir(K) -

1s)

(K - l,),

Fir(K)

- K,’

(94

386

APPROXIMATION

ALGORITHMS FOR PRODUCT-FORM NETWORKS

where (K - l,), is defined as in Eq. (9.2). Because the values Eir(K - 1,) are unknown, we are not able to compute the values Dirs(K) as well. Therefore we - estimate values for the difference D+,(K) and then we approximate the K,,(K - 1,) values by the following formula based on Eq. (9.4):

%r(K - 1s)= (K - L)r~ [Fir(K) + Dim(K)].

(9.5)

Note that if Dirs(K) = 0 we get the Bard Schweitzer approximation. The core algorithm of SCAT approximation needs the estimated values for Di,,(K) as input and, in addition, estimates for the mean values C,,.(K). The core algorithm can be described in the following three steps:

1STEP Cl 1 For i = 1,. . . , N and T, s = 1,. . . , R, compute with Eq. (9.5) the estimated values for the mean number xi, of jobs for the population (K - 1,). 1 STEP C2 1 Analyze the queueing network by using one iteration step of the MVA as in Step 3 of the Bard- Schweitzer algorithm. 1 STEP C3 ] Check the stopping condition. If: z;:)(K) max = i,r

- $-l’(K)1 KT

< E,

then stop the iteration, otherwise return to Step Cl (here the superscript cn) denotes the results of the nth iteration step), The use of & = (4000+ 16(KI)-l as a suitable value is suggested by [NeCh81]. The SCAT algorithm produces estimates of the differences DirS needed as input parameters of the core algorithm. So the core algorithm is run several times, and with the results of each run the differences Dir,(K) are estimated. The estimated differences Dirs(K) are computed by the SCAT algorithm, hence the name self-correcting approximation technique. To initialize the method, the jobs are distributed equally over all nodes of the network and the differences DirS (K) are set to zero. The SCAT algorithm can be described by the following four steps: m Use the core algorithm for the population vector K, and input values K+(K) = Kr/N and Dirs (K) = 0 for all i, r, s. The auxiliary values of the core algorithm are not changed by the SCAT algorithm. Use the core algorithm for each population vector (K - lj) for j = 1 * * 7R, with the input values Kir(K - lj) = (K - lj)r/N and Dirs(K 1;; = 0 for all i, r, s. ~ For all i = 1,. . . , N and r, s = 1,. . . , R, compute the estimated values for the Pi,(K) and &(K - l,), respectively, from Eq. (9.3) and the estimated values for the Dirs(K) from Eq. (9.4).

APPROXIMATIONS

BASED ON THE MVA

387

/%$$$$&I Use the core algorithm for the population vector K with values l?i7.(K) computed in Step 1, and values Dirs(K), computed in Step 3. Finally, compute all other performance measures using Eqs. (7.21)-(7.31). The memory requirement and computation time of the SCAT algorithm grows as O(N. R2) and produces, in general, very good results. Now, the algorithm is used on an example: Consider the central-server model of Example 8.3 again, but Example 9.2 now we use the SCAT algorithm to solve the network. L-1 Use the core algorithm with K = 6 jobs in the network and the input data ??i (K) = K/N = 1.5 and Di (K) = 0, for i = 1, . . . ,4. With these assumptions and E = 0.01, the usage of the core algorithm will produce the same results as the Bard-Schweitzer algorithm as in Example 9.1. Therefore, we use those results directly: T2(6)

= 0.573,

T3(6)

= 1.145,

T4(6)

= 1.240,

X2(6)

= 2.267,

3?3(6) = 2.267,

K4(6)

= 1.227.

?;1(6) = 0.024, X(6) = 9.896, K1(6)

[m]

= 0.239,

Use the core algorithm for (K - 1) = 5 jobs and the inputs 1?,(5) = = 0, for i = 1,. . . ,4.

5/N = 1.25 and Di(5) -1

- 1) from Eq. (9.5):

Compute the estimated values for the K,(K Ki(4)

=

4.

(i(5) --g-

=

1

for i = 1,...,4.

1 STEP C2 1 One step of MVA: T1(5)

= i

[l + K1(4)]

= 0.04,

= 10.417,

K2(5)

(1

izzl = 1.667,

K3(5)

T2(5) Z,(5)

= 1.667,

= 0.4,

T3(5)

= T1(5)X(5)el

E4(5)

= 0.8, = 0.417,

= 1.25.

Check the stopping condition: pTp(5)-Kp(5))

max i

5

=0166 L

The iteration must be started again with Step Cl.

>&.

T4(5)

= 1.2,

APPROXIMATION ALGORITHMS FOR PRODUCT-FORM NETWORKS

388

For the sake of brevity, we do not present all the other iterations but give the results after the last (fourth) iteration: T3(5) = 0.989,

T1(5) = 0.024, X(5) = 9.405,

T2(5) = 0.495,

K1(5) = 0.222,

x2(5) = 1.861, &(5)

T4(5) = 1.123,

= 1.861, x4(5) = 1.056.

Compute with Eq. (9.4) th e estimated values for the differences Di(K): El (5) E,(S) 0.222 0.239 4 7. 1o-3 = 5 - = - = , 5 6 ’ 6 DQ(6) = m5) --j- K2(6) = -5 7 10-3 6 ** ’ X3(5) E3(6) -5 7 10-3 D3(6) = 5 - = 6 ” ’

&(6)

D4(6) = x4(5) --g-

- li;4@) ~ 6

=

6 72

lo-3

”

*

Use the core algorithm for K = 6 jobs in the network with the Ki (K) values from Step 1 and the Di(K) values from Step 3 as inputs. STEP cl

Estimated values: %(6) K1(5) = 5 * 6 X3(5)

[I

+ Dl(6)

= 1.861,

1

= 0.222,

E,(5)

= 1.861,

E4(5) = 1.056.

One step of MVA: Tl(6) = i

[l + fT1(5)] = 0.024,

T2(6) = 0.572,

T3(6) = 1.144,

T4(6) = 1.234,

X(6) = 9.908, x1(6) = Fl(6)X(6)el

= 0.242,

E,(6)

= 2.267, F3(6) = 2.267,

K4(6) = 1.223. -1

Stopping condition:

IK36) - $“(6) 1 = 8.128. 1O-3 < &, max i 6 is fulfilled and, therefore the iteration stops and the preceding results are the final results from the SCAT algorithm. The exact results (MVA) are (see Example 8.3) : K1(6) = 0.244,

G(6)

= 2.261,

K3(6) = 2.261,

z(6)

= 1.234.

APPROXIMATIONS

BASED ON THE MVA

389

9.1.2.2 Multiple Server A/odes If we wish to analyze networks with multiple server nodes, then from the MVA analysis, Eq. (8.31), we need not only the l?;i, values but also the conditional marginal probabilities ni(j 1 K - 1,). Similarly, for the SCAT algorithm, we need estimates not only for the values Ki,(K - ls) but also for the probabilities ni(j 1 (K - 1,). In [NeCh81] a technique is suggested to estimate the marginal probabilities by putting the probability mass as close as possible to the mean number of jobs in the network. Thus, for example, if 7?i(K - 1) = 2.6, then the probability that there are two jobs at node i is estimated to be 7ri(2 1 K - 1) = 0.4, and the probability that there are three jobs at node i is estimated to be ri(3 1 K - 1) = 0.6. This condition can be described by the following formulae. We define: Ceilingi, = 1Ei(K

- 1,)1 ,

flOOrir = 1ITi(K - lr)]

,

(9.6) (9.7)

with:

and estimate: ri(floori, ~i(ceiling;, ri(j

1K - 1,) = ceilingi, - E(i (K - 1,), ) K - lT) = 1 - ri(floori, ) K - 1,), I K - 1,) = 0 for j < floori, and j > ceiling;,.

(9.8) W) (9.10)

Now we need to modify the core algorithm of SCAT to analyze networks with multiple server nodes. This modified core algorithm, which takes suitable estimates of Dirs (K) and xiv as inputs, can be described in the following steps: Cl 1 For - i = 1,. . . , N and r, s = 1,. . . , R, compute estimates for the mean values Ki, (K - ls) using Eq. (9.5). Compute using Eqs. (9.7)-(9.10), for each multiple server node i, the estimated values of the marginal probabilities r;(j 1K - l,),j = 0,. . . , (mi - 2) and r = 1,. . . , R. (-1 Analyze the queueing network using one step of the MVA via Eqs. (8.38), (8.39), and (8.41). [STEP]

Check the stopping condition: E;;)(K) max i,r

- ??,‘,“-l’(K) K-

/ < e.

If this condition is not fulfilled, then return to Step Cl.

390

APPROXIMATION ALGORITHMS FOR PRODUCT-FORM NETWORKS

The SCAT algorithm, which computes the necessary estimated input values for the differences Dirs(K) and mean values Kir(K) for the core algorithm, differs from the algorithm for networks with single server nodes only in the usage of the modified core algorithm, as follows:

all Dir,(K)

Use the modified core algorithm for the population vector K, where values are set to 0. Use the modified

core algorithm

for each population

vector

(K - lj). Compute the Fi,, and Dirs values from the previous results (see Eqs. (9.3) and (9.4)). Use the modified core algorithm with the computed F and D values for the population vector K. The closed network from Example 8.4 (Fig. 8.6) is now anaExample 9.3 lyzed with the SCAT algorithm. The network contains N = 4 nodes and K = 3 jobs. The network parameters are given in Table 9.3. The analysis of the network is carried in the following four steps: Table 9.3

Input parameters for the net-

work of Example 9.3 i

ea

1 2 3 4

1 0.5 0.5 1

l/Pi

mi

0.5 0.6 0.8 1

2 1 1 00

The modified core algorithm for K = 3 jobs in the network with the input parameters ??i (K) = K/N = 0.75 and Di (K) = 0 for i = 1, . . . ,4 is executed as follows: (ml

Estimate values for the xi(K Ki

E.;(2) = 2 * 3

(3)

=O.J

- 1) from Eq. (9.5): for i=

1,...,4.

Estimate values for x1(0 1K - 1). B ecause of floor1 = 1x1 (K - l)] = Q and ceiling, = floor1 + 1 = 1, we get: nr(O 12) = ceiling, - Kr(2)

= 0.5.

APPROXIMATIONS

I]

BASED ON THE MVA

391

One step of MVA: ml-2

%(3) = +--&

+ c (ml - j - 1). m(j 12) j=o

1 -t%(2) [

1

= pm1

ml - l)q(O

[1+X1(2)+( -

%(3) = ;

[l +

T4(3) = $ = 1.

X(3) =

4

) 2)] = 0.5,

Ts(3) = ;

= 0.9,

K2(2)]

1

[l +Ks(2)]

= l.2,

= 1.176,

3

C G(3) i=l

F,(3)

= T1(3)X(3)el

Ks(3) = 0.706, [STEP]

= 0.588,

&(3)

x2(3) = 0.529,

= 1.176.

Check the stopping condition: /fT,(l)(3) - FiO’(3)) max i

-1

= 0.142 > E = 0.01.

3

Estimate values for the Ki(K

- 1):

= 0.392,

K2(2) = 0.353, r,(2)

Ks(2) = 0.471,

= 0.784.

Estimate values for nr(0 1 K - 1). With floor-r = L??r(K - l)] ceiling, = floor1 + 1 = It, we get: 7rr(O ( 2) = ceiling, - Kr(2) -1

= Q and

= 0.608.

One step of MVA:

After three iteration steps, we get the following results for the mean number of jobs: x,(3)

= 0.603,

K2(3) = 0.480,

Ka(3) = 0.710,

17,(3) = 1.207.

The modified core algorithm for (K - 1) = 2 jobs in the network with the input parameters K,(2) = 2/N = 0.5 and Di(2) = 0 for i = 1,. . . ,4 is executed. After three iteration steps, for the mean number of jobs we have: 37,(2) = 0.430,

p2(2) = 0.296,

Ks(2) = 0.415,

x4(2) = 0.859.

392

APPROXIMATION ALGORITHMS FOR PRODUCT-FORM NETWORKS

Determine the estimates for the I$ and Di values using Eqs. (9.3) and (9.4)) respectively: K(3)

Ji(3)

= y-

= 0.201,

m) Pg2) = y--

h(3)

K(3) = -j-

= 0.160,

F2(2) = 172(2) 2 = 0 . 148,

F3(3)

E3(3) = 0.237, = y-

F4(3)

= 3

K4

(3)

= 0.402,

= 0 .215 )

F3(2) = K3P) 2

= 0 .208 ,

F4(2) = X4(2) 2

= 0 .430 ,

Therefore we get: D1(3) = Fl(2) - Fl(3) = 0.014, &(3) = &(2) - F2(3) = -0.012, D3(3) = F3(2) - 5’s(3) = -0.029, D4(3) = F4(2) - F4(3) = 0.027. Execute the modified core algorithm for K = 3 jobs in the network with values Ki(K) from Step 1 and values Di(K) from Step 3 as inputs. I]

Estimate values for the IIi(K Z(3) yj[ J&(2) = 0.415, r;;,(2) =2

+ D1(3)

- 1):

1

= 0.430,

z,(2)

= 0.296,

?;;4(2) = 0.859.

Estimate value for: floor1 = [Kr (2)J = 0,

ceiling1 = floor1 + 1 = I.,

slrr(O 12) = ceiling1 - ‘I?, (2) = 0.570.

After two iterations, results:

the modified core algorithm yields the following final

T1(3) = 0.5, X(3) = 1.224,

Tz(3) = 0.776,

x,(3)

z,(3)

= 0.612,

T3(3) = 1.122, T4(3) = 1,

= 0.475, x3(3) = 0.687, x4(3) = 1.224,

The exact results are (see Example 8.4): T1(3) = 0.512, X(3) = 1.217, KI(3)

T2(3) = 0.776,

T3(3) = 1.127, T4(3) = IL,

= 0.624, Kz(3) = 0.473, K3(3) = 0.686, x4(3) = 1.217.

APPROXIMATIONS

BASED ON THE MVA

393

9.1.2.3 Extended SCAT Algorithm The accuracy of the SCAT algorithm for networks with multiple server nodes depends not only on the estimates of the ?i;i, (K - 1 s) values, but also on the approximation of the condition al marginal probabilities 7ri(j ] K - 1,). Th e suggestion of [NeCh81], to distribute the whole probability mass only between the two values adjacent to the mean is not very accurate. If, for example, ??i(K- IT) = number of jobs K,(K-l,), 2.6, then the probability of having 0, 1, 4, and 5 jobs at the node cannot be neglected. Especially in the case when values ??i(K - 1,) are close to rni (number of identical service units at the ith node), the discrepancy can be large. In this case, all relevant probabilities are zero because the values ~(j ( K - 1,) f or computing the mean response time are determined for j = 0 * * 7(mi - 2) only. In [AkBo88 a] an improvement over this approximation of-the conditional probabilities is recommended. In Akyildiz and Belch’s SCAT scheme, the marginal probabilities are spread over the whole range . . . ,2 \Ki(K)] + 1. They define the following two functions, 0 * * 7K,(K-l,), the first function, PR, being a scaling function with:

PR(1) = Q PR(n) = pmPR(n - 1)

for n = 2,. . . , rn~x{~~},

where the authors give 45 and 0.7, respectively, as the optimal values for ctr and p. The second function W is a weighting function that is defined for all i = l,... , max(K,) as follows: W(O,O) = 1, W(i, j) = W(i - 1, j) - wti

- Lww~) 100

7

for j = 1,. . . ) (i _ 1)

7

i-l W(i,

i) = 1 - c W(6.i). j=O

The values of the weighting function W(i, j), j = 0,. . . ,5 are given in Table 9.4. Table 9.4

Values of the weighting function W

i

W(i, 0)

W(i, 1)

W(i, 2)

W(i, 3)

W(i, 4)

W(i, 5)

0 1 2 3 4 5

1.0 0.55 0.377 0.294 0.248 0.222

0.45 0.308 0.240 0.203 0.181

0.315 0.245 0.208 0.185

0.221 0.187 0.166

0.154 0.138

0.108

In addition to these two functions, we need the values of floor, ceiling, and maxval, where floor and ceiling are defined as in Eqs. (9.7) and (9.6) and

394

APPROXIMATION

maxval

is defined

ALGORITHMS FOR PRODUCT-FORM NETWORKS

as follows: maxvali,

= min(2 flooriT + 1, mi).

(9.11)

The term maxval describes the maximum variance of the probability mass. To compute the conditional probabilities, we divide the probability mass into (floori, + 1) pairs, where the sum of each of these pairs has the combined probability mass W(i, j). Formally this computation can be described as follows: l

For j III 0 are j jobs’at 7ri (floori,

.

. , floor+, compute the ith node:

- j ] K - 1,) = W(floorir,

l-dist

l

For j = ceiling;, ni(j

with l

the cond it ion al probabilities

upperval upperval

upperval

= ceiling;,

lowerval

= flooriT - 1-dist = j.

, . . . , maxvali,,

there

- Zi(K - lT) - lowerval

’ (9.12)

= flooriT - j, + 1-dist,

we have:

1K - lT) = W(fl oar;,, u-dist) u-dist

l-dist)

that

- ni(floori,

- u-dist

] K - l,), (9.13)

= j - ceiling+.

For j > maxvali,

we get: ri(j

1K - 1,) = 0.

This approximation assumes that the Ki(K - lr) values are smaller than KT/2. If any of these values is greater than (K, - 1)/2, then we must also make sure that the value of upperval is not outside the range 0, . . . , (Kr - 1). In this case upperval is set to (K, - 1). More details of this SCAT technique can be found in [AkBo88a].

Example 9.4 Consider the network shown in Fig. 9.1. To compute the marginal probabilities we apply the improved technique of [AkBo88a]. The service time of a job at the ith node, i = 1,2, is exponentially distributed with rates ~1 = 0.5 and ,Q = 1. The queueing disciplines are FCFS and IS, respectively. There are K = 3 jobs in the network and ei = e2 = 1. m Execute the modified and the input parameters Ei(K)

core algorithm for K = 3 jobs in the network = K/N = 1.5 and Di(K) = 0 for i = 1,2.

APPROXIMATIONS

Fig. 9.1

(sTEpc1]

BASED OAJ THE MVA

395

A model of a multiprocessor system.

Estimate values for the Ki(2), Eq. (9.5): Jw) =l i=12 Ki(2) = 2 * yj9 * -7 ( )

Estimate value for nr(0 1K - 1). Since: floor1 = LKr (2)J = IL, ceiling1 = floor1 + 1 = 2, maxvalr = min(2. floor1 + 1,mr) = 2, ~r(0 12) is determined with: l-dist = floor1 - 0 = IL, upperval = ceiling, + 1 = 3, lowerval = floor1 - 1 = 0, from Eq. (9.12): 7rl(O 12) = W(l,l) I/

* 3-W2)3

=rJ

* .

One iteration step of the MVA:

After four iterations, we get the following results: Kr(3)

= 2.187,

x2(3) = 0.813.

Apply the modified core algorithm for K = 2 jobs with the input parameters Ki(2) = 1 and Di(2) = 0, for i = 1,2. msT”Ep

Estimate values for the xi(l):

396

APPROXlMATiON

ALGORITHMS FOR PRODUCT-FORM NETWORKS

Estimate values for nr(0 1 1). With: floor1 = LX,(l)]

=Q,

ceiling1 = floor1 + 1 = 1,

maxvalr = min(2. floor1 + 1, ml) = 1, we determine: 1-dist = Q,

upperval = Il,

lowerval = 0,

and use these in Eq. (9.12) to compute: 7rl(O 1 1) = W(O,O) * l-ml) -1

=u

1

. .

One iteration step of the MVA:

After two iterations, we get the following results: 1?;1(2) = 1.333, -1

E,(3)

= 0.667.

Estimate the difference with Eq. (9.4): Dr(3) = Fr(2) - Fr(3) = -0.062,

02(3) = Fz(2) - Fz(3) = 0.062.

Apply the modified core algorithm

for K = 3 jobs in the net-

work: -1

Estimate values for the zi(2): ??r(2) = 1.333,

z2(2) = 0.667.

Estimate value for ~r(0 ) 2). With: floor1 = [;ilr (Z)] = 1,

ceiling, = 2,

maxvalr = 2,

we determine: 1-dist = 1,

upperval = 3,

lowerval = 0,

and, therefore: n1(0 12) = W(l,l)* 1-1

3 - F,(2)

One iteration step of the MVA:

= 0.25.

SUMMATION METHOD

397

Table 9.5 Storage requirement for convolution, MVA, Bard-Schweitzer, and SCAT Algorithms Storage Requirement

Methods Convolution

Oc&~

MVA

OW.

+ 1))

r&G-

Bard-Schweitzer

O(N . R)

SCAT

O(N . R2)

+ 1))

After two iterations, we get the final results: Tr(3) = 2.57,

TZ(3) = 1,

X(3) = 0.84,

x1(3)

= 2.16,

&(3)

= 0.84.

A comparison with the exact values shows the accuracy of this technique for this example: Ti(3)

= 2.45,

Tz(3) = 1,

X(3) = 0.87,

Er(3)

= 2.13,

K2(3) = 0.87.

If we use the SCAT algorithm from Section 9.1.2.2 we get: E(3)

= 3,

Tz(3) = 1,

X(3) = 0.75,

Er(3)

= 2.25,

111;2(3)= 0.75.

The storage requirement of SCAT is independent of the number of jobs (= O(N . R2)) but higher th an for the Bard-Schweitzer approximation, especially for systems with many job classes and many jobs. The storage requirement for SCAT is much less than those of MVA or convolution (see Table 9.5). The accuracy is much better for the SCAT algorithm (the average deviation from exact results is approximately 3%) than for the Bard-Schweitzer approximation and, therefore, in practice SCAT is used more often than MVA, BardSchweitzer, or convolution to solve product-form queueing networks. Problem 9.1 algorithms.

Solve Problem 7.3 with the Bard-Schweitzer and the SCAT

Problem 9.2 algorithms.

Solve Problem 7.4 with the SCAT and the extended SCAT

9.2

SUMMATION

METHOD

The summation method (SUM) is based on the simple idea that for each node of a network, the mean number of jobs at the node is a function of the

398

APPROXIMATION

throughput

ALGORITHMS FOR PRODUCT-FORM NETWORKS

of this node: (9.14)

For nodes with load-independent service rates, function fi has the following properties: l

Xi must be in the range 0 5 Xi 5 mipi. For IS-nodes we have 0 < Xi 5 K* pi 3 because Xi = Xi/pi and Ki 5 K.

l

fi(Xi) 5 f(xi + A&) for AX, > 0, that is, fi is a monotone nondecreasing function of Xi.

l

fi(0)

=O.

Monotonicity does not hold for nodes with general load-dependent service rates and must therefore be examined for each individual case. For multiple server nodes the monotonicity is maintained. To analyze product-form queueing networks, [BFS87] suggest the following formulae: t

Pi 1-K-l

Type-1,2,4 (mi = l),

’ KPi

fi

(Xi)

=

Ki

=

{

T?2ifli

Pi

+

AL,,

l-K-mi-l

K -mi

Type-l

(mi > l),

pi

-Ai \ pi ’

Type-3. (9.15)

The utilization

pi is computed using Eq. (7.21) and the waiting probability with Eq. (6.80). It should be noted that Eq. (9.15) gives exact values only for Type-3 nodes while the equations for the other node types are approximations. The approximate equations for Type-l (-/M/l), Type-2, and Type-4 nodes can be derived from Eq. (6.13), Ki = pi/(1 - pi), and the introduction of a correction factor (K - 1)/K, respectively. The correction factor is motivated by the fact -that in the case of JO s are at node i, hence, for pi = 1 we have Ki = K. The idea pi=l,allK’b of the correction factor can also be found in the equation for Type-l (-/M/m) nodes by considering Eq. (6.29). If we assume that the functions fi are given for all nodes i of a closed network, then the following equation for the whole network can be obtained by summing over all functions fi of the nodes in the network: Pm2 with Eq. (6.28) or approximately

CRi i=l

= C i=l

fi(Xi)

= K.

(9.16)

399

SUMMATION METHOD

With the relation A; = A. ei, an equation to determine the overall throughput X of the system can be given as: 2

(9.17)

fi(X+ ei) = g(X) = K.

i=l

In the multiple class case, the Function (9.14) can be extended:

Kir = fir(b).

(9.18)

The function fir ( Air) has the following characteristics: l

l

fir is defined in the range 0 2 Ai, 2 pi, . rni and, in case of an infinite server node, 0 5 Ai, 5 pir . K,. fir is a monotone non decreasing function. fir (Air) 5 fir (Ai, + e), e > 0.

. fir(O) = 0. For multiclass product-form queue&g networks, the following formulae, as extensions of Eq. (9.15), are given: Pir

l--

K-l K

Pi

Type-1,2,4 (mi = 1)

’ Pir

. P,,,

l-K-mi-l K -mi

Type-l

(mi > 1),

(9.19)

Pi Type-3,

with: Pi

=

(9.20)

5Pir7

r=l R

(9.21)

K=xK,.

For the probability of waiting P,,, we can use the same formula as for the single class case (Eq. (6.28)) and Eq. (9.20) for the utilization pi. Now we obtain a system of equations for the throughputs A, by summing over all functions fir:

5K;r

i=l

=

5

fir(Air)

i=l

=

Kr

(T

=

I,,

.

l

,

R).

(9.22)

400

APPROXIMATION

ALGORITHMS FOR PRODUCT-FORM NETWORKS

With Xi, = Xreir we obtain the system of equations:

fJi.(&-ei,)=g&b)=K,

r = l,...,R.

(9.23)

i=l

In the next section, we introduce solution algorithms to determine the throughput X in the single class case or the throughputs X, in the multiple class case using the preceding fixed-point equations. 9.2.1

Single

Class Networks

An algorithm for single class queueing networks to determine X with the help of Eq. (9.17) and the monotonicity of fi (Xi) has the following form: m-!

Initialization.

Choose Xl = 0 as a lower bound for the throughput as an upper bound. For -/G/co-nodes,

1-1

rni must be

Use the bisection technique to determine X:

STEP 2 1 CZYI

Xl + L Set x = 2 *

-1

Determine g(X) = 2 fi (X . e,,) , w h ere the functions f; (X. ei) are i=l determined by using Eqs. (9.15) and (9.14). -1 If g(X) = K f e, then stop the iteration and compute the performance measures of the network with the equations from Section 7.1. If g(X) > K, set X, = X and go back to Step 2.1. If g(X) < K, set Xl = X and go also back to Step 2.1. The advantage of this method is that the performance measures can be computed very easily. Furthermore, if we use suitable functions fi, then the number K of jobs in the network and the number of servers rni does not affect the computation time. In the case of monotonicity, the convergence is always guaranteed. By using suitable functions fi (Xi), this method can also be used for an approximate analysis of closed non-product-form networks (see Section 10.1.4). In [BFS87] and [AkBoBBb], it is shown that the summation method is also well suited for solving optimization problems in queueing networks (see Chapter 11). In the following example we show the use of the summation method: Consider Example 8.4 from Section 8.2.1. This network Example 9.5 is examined again to have a comparison between the exact values and the approximate ones as computed with the summation method. The network

SUMMATION METHOD

Table 9.6 Input parameters work of Example 9.5 i

e,

1 2 3 4

1 0.5 0.5 1

401

for the net-

mi

lIPi

2 1 1 cm

0.5 0.6 0.8 1

parameters are given in Table 9.6. The analysis of the network is carried out in the following steps:

Initialization: Xl = Q and

A, = min i

Bisection: Xl + A, X = 2=

STEP 2 1 r---Y-y

1.25.

[STEP/ Computation of the functions fi(Xi), For the -/M/2 node 1 we have: p1 =

~J-1 = 0.3125 kw ml

i = 1,2,3,4 with Eq. (9.15).

and Pm1 = 0.149

Eq. (6.28),

and, therefore: f&Q

= 2p1 + PIP,,

= 0.672.

Furthermore: fzP2)

= 0.5

= fi

with p2 = 0.375,

3 f3P3)

=

= 0.75

&

with p3 = 0.5,

3 f4(X4)

=

;

= 1.25.

Thus, we get: g(X) = 5

fi(Xi)

= 3.172.

i=l

l-2.3] Check the stopping condition. A, = X = 1.25 and go on with the iteration.

Because of g(X) > K, we set

402

APPROXIMATION

ALGORITHMS FOR PRODUCT-FORM NETWORKS

[STEP x

fr(xr> = 2Pr + Pi&, f$i2) = 0.214 f3(X3)

= 0.3

f&b)

= 0.625.

=

AZ

-

+

AA

2

= 0.319

= 0.625.

with pi = 0.156 and Pm1 = 0.042, with p2 = 0.1875, with pa = 0.25,

It follows that: g(X) = -&(Xi)

= 1.458.

i=l

-1

Because of g(X) < K, we set Xl = X = 0.625.

x

=

-= AZ

+

0.625 + 1.25 = o g375 -2

LL

2

This iteration is continued until the value of g(X) equals (within E = 0.001) the number of jobs K in the system. To make the bisection clearer, we show the series of Xl and X, values in Table 9.7. Table 9.7

Intervals for Example 9.5

Step:

0

1

2

3

4

5

6

7

8

9

10

h Lt

0 2.5

0 1.25

0.625 1.25

0.9375 1.25

1.094 1.25

1.172 1.25

1.172 1.211

1.191 1.211

1.191 1.201

1.191 1.196

1.191 1.194

The overall throughput of the network is X = 1.193 as computed by this approximation method. Compare this with the exact value of the overall throughput is X = 1.217 The approximate values of the throughputs and the mean number of jobs calculated using the SUM (Eq. (9.14)) and the exact values for this network calculated using MVA (see Example 8.4) are summarized in Table 9.8.

SUMMATION ME T/-l00

403

Table 9.8 Exact and approximate values for the throughputs Xi and the number of jobs K, for Example 9.5

i

xZSUM &MVA

K K ZMVA

-ZSUM

9.2.2

Multiclass

1

2

3

4

1.193 1.218 0.637 0.624

0.596 0.609 0.470 0.473

0.596 0.609 0.700 0.686

1.193 1.218 1.193 1.217

Networks

Now we assume R job classes and K = (Ki, K2, . . . , KR) jobs in the network. Therefore, we get a system of R equations. In general, g, are non-linear functions of A,: gT(Xi, X2,. . . , AR). This is a coupled system of non-linear equations in the throughput of the R job classes and, hence, it is not possible to solve each equation separately from the others. We first transform the system of equations into the fixed-point form:

x = f(x). Under certain conditions f has only one fixed-point and can then be used to calculate x iteratively using successive substitution as follows:

This method of solution is also called fixed-point iteration. In the case of the SUM, for the calculation of the throughputs A,, we get the following system of equations: ~~=f$-(A,.ei,)

= K,,

T= l,...,

R.

(9.24)

i=l

i=l

To apply the fixed-point iteration, this equation can easily be transformed to:

or: Jb=

N

K?-

C fixi&b

see [BaTh94].

= fr(h, . ei,)

* * JR)

= h-(X),

(9.25)

404

APPROXlMATlON

ALGORITHMS FOR PRODUCT-FORM NETWORKS

For the functions fixiT, we get from Eq. (9.25):

,

ei, Pir

I-K-1K

eir mi *j&r

fiXir = { eir

Type-1,2,4 m; = 1,

. Pi ’

I

(9.26)

l-K-mi-l K - mi

/Jir

*

% (Pi>, Type-l

‘pi

ei,

\b t%r

Type-37 ’

and the throughputs ~~

rni > 1,

A, can be obtained by the following iteration:

Initialization: Xl = x2 = * * * = AR = 0.00001; Compute the throughput

E = 0.00001.

A, for all r = 1, . . . , R:

Determine error norm:

e=

C( k2 - h-n+l)2’ 11r=l

If e > E, go to Step 2. ~

Compute other performance measures.

The Newton-Raphson method can also be used to solve the non-linear system of equations for the SUM [HahnSO]. There is no significant difference between fixed-point iteration and the Newton-Raphson method in our experience. The computing time is tolerable for small networks for both methods (below 1 second), but it increases dramatically for the Newton-Raphson method (up to several hours) while it is still tolerable for the fixed-point iteration (several seconds) [BaTh94], Therefore, the fixed-point iteration should preferably be used. The relative error can be computed with the formula: difference =

(exact value - computed value]. 1oo 7 exact value

BOTTAPPRO>( METHOD

405

and gives, for the previous example, an error of 2.6% for the overall throughput. Experiments have shown that the relative error for the SUM method is between 5% and 15%. It is clear that the results can be further improved if better functions fi could be found. Problem

9.3

9.3

Solve Problem 7.3 and Problem 7.4 using the SUM.

BOTTAPPROX

METHOD

We have seen that, compared to the MVA, the SUM approximation needs much less memory and computation time. But if we take a closer look at it, we will see that the number of iterations to obtain reasonably accurate results is still very large for many networks. Now the question arises whether it is possible to choose the initial guess of throughput X(O) so that the number of iterations can be reduced while still maintaining the accuracy of the results. A good initial value of throughput based on the indices of the bottleneck node is suggested by [BoFi93]. Recall that the node with the highest utilization pi is called the bottleneck node. The performance measures of the bottleneck node are characterized by the index bott. The resulting approximation method is called bottapprox (BOTT). 9.3.1

Initial Value of X

For the throughput

X we have the relation: Pi * mi A= -*Pi,

i = l,...,N,

ei

with: 0 5 pi < 1,

i = l,...,

N.

In contrast to the SUM in the BOTT we assume that the utilization &,tt of the bottleneck node is very close to 1, and therefore we can choose for the initial value of the throughput X(O):

where argmin is the function that returns the index of the node with the minimum value. 9.3.2

Single Class Networks

The BOTT is an iterative method where for each node i in each iteration step, the mean number of jobs Ki is determined as a function of the chosen

APPROXIMATION

406

throughput

ALGORITHMS FOR PRODUCT-FORM NETWORKS

A!“): (9.27)

?;;i = f@i).

For the function fi, the same approximations are used as in the SUM. By summing over all nodes, we get for the function g(X) : (9.28) of jobs K in the

This sum g(X) is compared to the correction factor: K

and a

(9.29)

corr = SC4 ’

is determined. We assume th .at in each node i the error in computing the mean number of jobs K; is the same, and therefore for each node the following relation holds: (9.30) For the bottleneck node the new mean number of jobs in the node is computed as follows: jp+1) bott

and the new utilization

= jp) .- K bott g(X) ’

(9.31)

(k+l) can then be determined as a function of the Pbott

new mean number of jobs -iTfoTtl’ at the bottleneck node using Eq. (9.15), that is: (k+l) Pbott

=

qptl’)*

(9.32)

Now the new value for the throughput X at the bottleneck node can be computed as a function of the new utilization (Eqs. (7.23) and (7.25)): A(‘“+l)

_ -

(k+l)

Pbott

* mbott

’ pbott

.

(9.33)

ebott

If none of the following stopping conditions is fulfilled then the iteration repeated: l

is

If the correction factor corr is in the interval [(l - c), . . . , (1 + E)], then stop the iteration. Tests have shown that for E = 0.025 we get good results. For smaller E, only the number of iterations increased but the accuracy of the results did not improve significantly.

BOTTAPPROX METHOD

407

l

If in the computation the value of Pb&t = h(??b,tt) is greater than 1, then set P&t = 1 and stop the iteration.

l

To prevent endless loops, the procedure is stopped when the number of iterations reaches a predefined upper limit.

The function h previously mentioned for different node types is defined as follows (see Eq. (9.15)): 1. For node types -/M/l-FCFS,

-/G/l-PS,

h(Kbott) =

and -/G/l-LCFS-PR: xbott

2. For node type -/G/oo-IS: xbott

h(%ott ) = 7.

3. For node type -/M/m-FCFS it is not possible to give an exact expression for fi and therefore an appropriate approximation is chosen (see [BoFi93]):

2-f

- mbott

’

with: f=

K

-

-

1 ?

K

b = Kbott C

mbott -

mbott . f + mbott

-I- Pmbott,

= b2 - 4 * mbott * Kbott . f.

Now we give an algorithm for the BOTT: Initialize:

Iteration: [d

Compute the following performance measures for i = 1, . . . , N:

408

APPROXIMATION ALGORITHMS FOR PRODUCT-FORM NETWORKS

-1

Compute the new throughput

g(X)= &,

A:

Kbott=Ebott*-$jY

i=l

x = Pbott . mbott

Pbott = @bott),

[I

. pbott

ebott

Check the stopping condition:

I I K g(X)

< E*

If it is fulfilled, then stop the iteration, otherwise return to Step 2.1. For i = 1, . . . , N compute:

To estimate the accuracy of the BOTT and to compare it with the SUM, we investigated 64 different product-form networks. The mean deviation from the exact solution for the overall throughput X and for the mean response time T was 2%. The corresponding value for the summation method was about 3%. The number of iterations of the SUM were about three times the number of iterations of the BOTT. For details see Table 9.9. Table 9.9

Deviation in percent (%) f rom the exact values (obtained by MVA) Error in X

min mean max

9.3.3

Error in T

Number of Iterations

SUM

BOTT

SUM

BOTT

SUM

BOTT

0 2.60 7.40

0 2.10 13.90

0 2.70 7.80

0 2.10 12.20

1 6.60 13

1 2.30 5

Multiclass

Networks

One of the advantages of the BOTT method is that it can easily be extended to multiple class networks. The necessary function h,(zi,) is given by (Eq. (9.19)): 1. For node types -/M/l-FCFS, hr(%ott,r)

-/G/l-PS

= Kbott,r

’

and -/G/l-PR: 1 - 7

’ pbott)

.

(9.34)

409

BOTTAPPROX METHOD

2. For node type -/G/oo-IS,

/L,(K~,~): WKbott,r

) =

Fbott K.

r

(9.35)

3. For the node type -/M/m-FCFS: h-(Kbott,r)

Ebott,r

=

1

I mbott

’ Pmbott

+

1 _

K

K

mbott -

-

1

(Pbott

)

’ Pbott

mbott

(9.36) Remark: Eqs. (9.34) and (9.36) are approximations because we used &ott,r part of &Ott = c,r”=, Pbott,i when we created the h,(Ki,,) function. Now we give the algorithm for the multiple class BOTT: Initialize:

,s.,,Nj{e}, A-= iEIyin

forr=L..,R,

bott, r = argmin {y},r. Iteration: IsTEh,

Compute the performance measures: Air

= b%-,

pir

=

Air

zr

mi

* j&r

= f&h,

’ * * a, AR),

r = l,...,R, i = l,...,N.

Pi=~Pir7 r=l

-1

Compute the new throughputs: N g(Xr)

= CKir, i=l

Ebott,r

= Ebott,r

Pbott,r xr

KYS(h)

=

h- (Ebott,r),

=

Pbott,r

* mbott

b

’

* pbott,r ,

eb0tt,7-

r = l,...,R.

410

APPROXIMATION ALGORITHMS FOR PRODUCT-FORM NETWORKS

-1

Stop the iteration for class T. If:

go to Step 2.1, else stop iteration for class r. Final value of Ki,:

Ki, = Fir . - KT dM

’

r=l,...,

R, i=l,...,

N.

The deviation from the exact results is similar to the one in the single class case for the SUM and is slightly worse but tolerable for the BOTT (see Table 9.10). The average number of iterations for the examples of Table 9.10 Tab/e 9.10 Deviation in percent (%) from the exact results for the throughput X (average of 56 examples) BOTT min mean max

SUM

0

0

6.7 38.62

4.68 40.02

used by the bottapprox algorithm is 4.5, and the maximal value is 20, which is much smaller in comparison with the SUM. Problem

9.4

9.4

Solve Problem 7.3 and Problem 7.4 using the BOTT.

BOUNDS ANALYSIS

Another possibility for reducing the computation time for the analysis of product-form queueing networks is to derive the upper and lower bounds for the performance measures instead of the exact values. In this section we introduce two methods for computing upper and lower bounds, asymptotic bounds analysis (ABA) and balanced-job- bounds analysis (B JB). These two methods use simple equations to determine the bounds. Bounds for the system throughput and for the mean number of jobs are calculated. The largest possible throughput and the lowest possible response time of the network are called optimistic bounds, and the lowest possible throughput and greatest possible mean response time are called pessimistic bounds. So we have:

BOUNDS ANALYSIS

411

With these relations other bounds for the performance measures such as throughputs and utilizations of the nodes can be determined. In the following we make a distinction between three different network types: l

Type A describes a closed network containing no infinite server nodes.

l

Type B describes a closed network with IS nodes.

l

Type C describes any kind of open network.

We restrict our consideration to networks with only one job class. Generalizations to networks with several job classes can be found in [EaSe86]. 9.4.1

Asymptotic

Bounds Analysis

The ABA [MuWo74a, Klei76, DeBu78] gives upper bounds for the system throughput and lower bounds for the mean response time of a queueing model (optimistic bounds). The only condition is that the service rates have to be independent of the number of jobs at the node or in the network. As inputs for the ABA method, the maximum relative utilization of all non-IS-nodes, 2 miLX,and the sum xsum of the relative utilizations of these nodes are needed:

X max

--

max(xi) i

and

xs,,

= c

xi,

where xi = ei/pi is the relative utilization of node i (Eq. (7.6)). The relative utilization of an IS node (thinking time at the terminals) is called 2. At first we consider the case of an open network (Type C): Since the utilization pi = Xi/pi with Xi = X. ei, the following relation holds: p; = A* x;.

(9.37)

Due to the fact that no node can have a utilization greater than 1, the upper bound for the throughput of an open network cannot exceed Xsat = l/xmax if the network is to remain stable. Therefore, Xsat is the lowest arrival rate where one of the nodes in the system is fully utilized. The node that is utilized most is the node with the highest relative utilization xrnax. For this bottleneck node we have pmax = X. x max 5 1, and the optimistic bound for the throughput is therefore: (9.38) To determine the optimistic bounds for the mean response time, we assume that in the best case no job in the network is hindered by another job,. With this assumption, the waiting time of a job is zero. Because the relative utilization xi = ei/pi of a node is the mean time a job spends being served at

412

APPROXIMATION

ALGORITHMS FOR PRODUCT-FORM NETWORKS

this node, the mean system response time in the non-IS nodes is simply given as the sum of the relative utilizations. Therefore, for the optimistic bounds on mean response time we get:

For closed networks we consider networks with IS nodes (Type B). From the results obtained for such networks we can easily derive those for networks of Type A by setting the mean think time 2 at the terminal nodes to zero. The bounds are determined by examining the network behavior under both light and heavy loads. We start with the assumption of a heavily loaded system (many jobs are in the system). The greater the number of jobs K in the network, the higher the utilization of the individual nodes. Since:

Pi(K)= qqxi I 1, the highest possible overall network throughput is restricted by each node in the network, especially by the bottleneck node. As in open models we have:

X(K) 5 1. Xmax

Now assume that there are only a few jobs in the network (lightly loaded case). In the extreme case K = 1, the network throughput is given by +Z). For K = 2,3,... jobs in the network, the throughput reaches l/(Xsum its maximum when each job in the system is not hindered by other jobs. In this case we have X(K) = K/(xsum + 2). These observations can be summarized in the following upper bound for the network throughput : (9.40) To determine the bound for the mean response time F(K), Eq. (9.40) with the help of Little’s theorem: K T(K)

we transform

1

< min + 2 C Xmax

‘2

and get:

(9.41) or

max {xsum, K*xmax - 2) I T(K)*

The asymptotic bounds for all three network types are listed in Table 9.11.

BOUNDS ANALYSIS

Table 9.11 Network

413

Summary of the ABA bounds

Type

ABA

Bounds

A

x

B

Xl1

C

xmax

T

T(K)

2 max { xsurn

) K*

xmax

-

2)

As an example for the use of ABA, consider the closed Example 9.6 product-form queueing network given in Fig. 9.2. There are K = 20 jobs in the network and the mean service times and the visit ratios are given as follows: = 4.6, el = 2,

~/PI

Fig. 9.2

~/,cL~ = 8, e2 = e3 = 1.

l/p3

Single class queueing

= 120 = 2,

network.

At first we determine: 2 max

=max

el

--,--{ Pl

e2

= 9.2,

x,,,

= 3Al + ;

= 17.2,

Pa >

With these values and the formulae asymptotic bounds immediately.

given in Table 9.11, we can determine

the

414

APPROXIMATION

ALGORITHMS FOR PRODUCT-FORM NETWORKS

Throughput: K

X(K) 5 min {

xsurn

‘> + 2’

=min{-$&;$-}

=0.109.

xmax

Mean response time: T(K)

2 max {CC,,,, PC.xmax - 2) = max(17.2; 64) = 64.

Now the bounds for the other performance measures can be computed. With the formula pi = X. xi, we can, for example, compute the upper bounds for the utilizations pr 5 1 and p2 < 0.87. For this example, the exact values for throughput and response time are X(K) = 0.100 and T(K) = 80.28, respectively. 9.4.2

Balanced

Job Bounds

Analysis

With the BJB method [ZSEG82], upper as well as lower bounds can be obtained for the system throughput and mean response time of product-form networks. In general, using this analysis we get better results than with the ABA method. The derivation of bounds by the BJB method is demonstrated only for closed networks of Type A. The derivation of the bounds for the other two network types is similar and is therefore left to the reader as an exercise. In the derivation of BJB, it is assumed that the network is balanced: that is, the relative utilizations of all nodes are the same: xi = . . . = xN = x. The throughput of a balanced network can be computed by applying Little’s theorem:

X(K)= N K C Ti(K)’ i=l where Fi (K) is the mean response time in the ith node with K jobs in the network. For product-form networks, Reiser and Lavenberg [ReLa80] have shown that: Ti(K)

= [I + K,(K

- I)] .x.

If we combine the two preceding formulae, we get: K X(K)

=

N

C [l +Ki(K

- I)] .x

= (N+ff-

l).x’

(9.42)

i=l

Equation (9.42) can be used to determine the throughput bounds of any type of product-form queueing network. For this computation, let xmin and xrnax,

BOUNDS ANALYSIS

415

respectively, denote the minimum and maximum relative utilization of an individual node in the network. The basic idea of the BJB analysis is to enclose the considered network by two adjoining balanced networks: 1. An “optimistic” network, where the same relative utilization assumed for all nodes,

x,i,

is

2. A “pessimistic” network, where the same relative utilization assumed for all nodes.

x,,,

is

With Eq. (9.42), the BJB bounds for the throughputs can be determined as:

(N +“K -1).-xi,,LX(K) L(N +“K -1).-Xii,’

(9.43)

Similarly, the bounds for the mean response time can be determined by using Little’s theorem: (N+K-1)*x

min 5 T(K)

5 (N + K - l)‘Xmax-

(9.44)

The optimistic bounds can be improved by observing that among all the networks with relative overall utilization xsum, the one with the relative utilization, xi = xsUrn/N for all i, has the highest throughput. If xave = xsum/N denotes the average relative utilization of the individual nodes in the network, then the improved optimistic bounds are given by:

X(K) L

K .-= 1 K + N - 1 x:,,,

K (9.45) Xsum

+ (K

-

l)xa,,

and: Xsum

+ (K

-

1)xaw

(9.46)

T(K),

5

respectively. The optimistic network, the network with the highest throughput, has, therefore, N = xsum/xave nodes with relative utilization x,,. Analogously, among all networks with relative overall utilization x,,, and maximum relative utilization xmax, the network with the lowest throughput has altogether xsum/xmax nodes with relative overall utilization x,,,. Therefore the improved pessimistic bounds are given by: K K+z-l

.-= 1 xmax

K xsum

+ (K

I X(K) -

(9.47)

1)xmax

and:

T(K) L Gum respectively.

+ (K

-

l)Xmax,

(9.48)

416

APPROXIMATION ALGORITHMS FOR PRODUCT-FORM NETWORKS

Table 9.12

Summary of the BJB bounds [LZGS84]

Network Type A

T

B

BJB Bounds K

K Xsum + (K - l)xmax

’ X(K) ’

+ CK - ljxave

< gicK) I

xsum

l+&

1 -

Zsum

+ (K

-

1)~s~

+ (K - l)xmax 1+*

Gum

C

Zsum

IT<

XXave

sum

zsum l-

AXmax

The inequalities for open networks can be obtained in a similar way. In Table 9.12, the BJB bounds for all three network types are summarized. The equations for the two closed network types are identical, if the value of 2 is set to zero. Consider the network of Example 9.6 again, but now we Example 9.7 determine the bounds with the BJB method. With x,,~ = x,,,/N = 8.6, we can obtain the bounds directly from Table 9.12. Throughput: 20 20 < X(K) I 17.2 + 120 + 129.5 17.2 + 120 + 20.5’ 0.075 < X(K) < 0.127. Response time: 174.8 17.2 + = < ‘?;(K) 5 17.2 + 1349’ 7.977 37.70 < T(K) < 146.8. A comparison of the results of both methods shows that for this example we get better results with the ABA method than with the BJB method. Therefore the conclusion is irresistible that using both methods in combination will reduce the area where the performance measures can lie. Figures 9.3 and 9.4 show the results of the ABA and BJB values for the throughput and mean response time as functions of the number of jobs in the network, respectively. The point of intersection for the two ABA bounds is given by: Jq* = 2sum +Z . Xmax

BOUNDS ANALYSIS

417

From the point: (Xsum

K+ =

+ 2)2

(X sum + 2).

-

Xmax

Xsumxave -

Xsumxave

upwards, the optimistic curve of the ABA method gives better results than the B JB one. These bounds say that the exact values for the throughput and the mean response time reside in the shaded area.

BJ&,t ABA

B JB,,,

I I

Y

1

Fig. 9.3

K+

ABA and BJB throughput

A

I

I I

1

bounds as a function of K.

ABA

B J&w,,

Fig. 9.4

-,K

,

K*

I I

K*

1

B JB,,t

;-K

K+

ABA and BJB response time bounds as a function of K.

The performance bound hierarchy method for determing the bounds of the performance measures was developed by [EaSe83]. In this method we have a hierarchy of upper and lower bounds for the throughput. These bounds converge to the exact solution of the network. The method is based on the

418

APPROXIMATION ALGORITHMS FOR PRODUCT-FORM NETWORKS

MVA. Thus, there is a direct relation between the accuracy of the bounds and the cost of computation. If the number of iterations is high, then the cost is higher but the bounds are closer together. On the other hand, when the number of iterations is lower, the cost is lower but the bounds are farther apart. The extension of this method to multiclass networks can be found in [EaSe86]. 0th er versions can be found in [McMi84], [Kero86], [HsLa87], and [ShYaSS]. Problem 9.5 Solve Problem 7.3 with the ABA and BJB methods and compare the results.

9.5

NETWORKS

WITH VARIABILITIES

IN WORKLOAD

Performance models need values of input parameters in order to predict values of performance measures. In practice, input parameters are estimated from measurements on real systems. Thus each input parameter will have a confidence interval in addition to a point estimate. Traditional algorithms such as MVA or SCAT accept only single values as input parameters and compute single values for the performance measures. In this section we describe a method that allows intervals as input parameters. As a result, the predicted performance measures are also given as intervals. The treatment here is adapted from [HLM96] and [MaRa95]. We can describe the workload variabilities, e.g., service rate variabilities, by intervals: I = [i-,if].

(9.49)

If the performance measures are monotone with respect to the input parameters, given by intervals, then a simple interval based algorithm can be derived from a given single value algorithm y = y (~1, ~2, . . . , pi), where y is the considered performance measure and the service rates are given by intervals: Ipn = [p,,pL]

n=

l,,..,

N.

Now the interval based algorithm can be described as follows: Calculate the performance measures y for the lower and upper bounds of the service rates:

!I- =Y(/q?.**&A

(9.51)

!I+ = Y(/& - * - 7P$).

(9.52)

Calculate the intervals for the performance measure y . If y is monotonically increasing in the parameter, then:

Iy = [Y-4+1.

(9.53)

NETWORKS WITH VARIABILITIES IN WORKLOAD

419

If y is monotonically decreasing, then: (9.54)

Iy = [Y+,Y-l.

Example 9.8 Consider a closed queueing network with two nodes and er = e2 = 1, K = 5, and the two intervals: Ipl = [l/

set, 2/ set]

Ip2 = [3/ set, 4/ set].

(9.55)

The interesting performance measures are the throughput Xr = X2 = X, which is monotonically increasing with service rates and the mean response time T, which is monotonically decreasing with service rates.

A- = X(1/ set, 3/ set) = 0.99/ set,

(9.56)

set) = 1.97/ set,

(9.57)

A+ = x(2/

set, 4/

T- = T( l/ set, 3/ set) = 5.01 set,

(9.58)

T+

(9.59)

= T(2/sec,4/sec)

Interval for the throughput

= 2.54sec.

X:

1~ = [A-, A+] = [0.99/ set, 1.91/ set].

(9.60)

Interval for the mean response time T: IT = [T+,T-]

1 Workstations

Ethernet

Fig. 9.5

= [2.54sec, 5.01 set].

(9.61)

1

Model of the client-server system.

As another example, we consider a client-server system where Example 9.9 the client workstations (labeled as node 1) submit SQL requests to a server, consisting of a CPU and n disk devices (labeled as nodes 3 and 4) [HLM96]. The workstations are modeled as -/G/l-IS station, while the CPU and disk device are modeled as -/M/l-FCFS stations. The ethernet (labeled as node 2), which connects the workstations with the server, can approximately be modeled by a -/G/l-IS server. The resulting network is shown in Fig. 9.5. The parameters of the model are listed in Table 9.13. Note that workstation service rate is specified as three intervals each having a given probability. For further details see [HLM96].

APPROXIMATION ALGORITHMS FOR PRODUCT-FORM NETWORKS

420

Parameters

Table 9.13 Node

I1 = [2.3,2.7] : p(Il) = 0.2 I2 = [7.4,8.4] : ~(12) = 0.5 13 = [16,21] : p(Iz) = 0.3 0.00185 set 0.12 set 0.054 set

Ethernet CPU Disk

system model Visit Ratios

Service Time

Workstation

9.6

of the client-server

1.0 2.0 1.0 1.0

SUMMARY

The main advantages and disadvantages of the approximation algorithms for product-form queueing networks described in this chapter are summarized in Table 9.14. Tab/e 9.14 networks.

Comparison

of the approximation

algorithms

queueing

Disadvantages

Advantages

Algorithms

for product-form

Very low storage requirement

SCAT

Good accuracy Very low storage requirement compared with MVA or convolution

Needs more iterations

SUM

Easy ment Low ment Easy form

Accuracy is not very high (but sufficient for most applications)

to understand

and

time

No multiple server nodes Low accuracy

Bard-Schweitzer CBS)

and imple-

than BS

storage and time requireto extend to non-productnetworks

BOTT

The same advantages as SUM Fewer iterations than SUM For multiple class networks, easier to implement than SUM Accuracy slightly better than SUM

Accuracy is not very high (but sufficient for most applications)

ABA, BJB

Well suited for a bottleneck analysis In the design phase, to obtain a rough prediction (insight, understanding) of the performance of a system Extremely low storage and time requirement

Only for single class networks Only upper and lower bounds

IO Algorithms for Non-Product-Form Networks

Although many algorithms are available for solving product-form queueing networks (see Chapters 8 and 9), most practical queueing problems lead to non-product-form networks. If the network is Markovian (or can be Markovized), automated generation and solution of the underlying CTMC via stochastic Petri nets (SPNs) is an option provided the number of states is fewer than a million. Instead of the costly alternative of a discrete-event simulation, approximate solution may be considered. Many approximation methods for non-product-form networks are discussed in this chapter. These algorithms and corresponding sections of this chapter are laid out as shown in the flowchart of Fig. 10.1. Networks with non-exponentially distributed service times are treated in the next section, while networks with FCFS nodes having different service times for different classes are treated in Section 10.2. Priority queueing networks and networks with simultaneous resource possession are treated in Sections 10.3 and 10.4, respectively. Network models of programs with internal concurrency are treated in Section 10.5. Fork-join systems and parallel processing are treated in Section 10.6. Networks with asymmetric server nodes and blocking networks are discussed in Section 10.7 and Section 10.8, respectively. 421

422

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

Fig. 10.1 Different algorithms for non-product-form tions.

networks and corresponding sec-

NON-EXPONENTIAL

10.1 10.1.1

NON-EXPONENTIAL Diffusion

DISTRIBUTIONS

423

DISTRIBUTIONS

Approximation

Although we present diffusion approximation as a method to deal with networks with non-exponential service time and interarrival time distributions, it is possible to apply this technique to Markovian networks as well. The diffusion approximation is a technique based on an approximate productform solution. In this approximation, the discrete state stochastic process {K(t) I t 2 O> d escribing the number of jobs at the ith node at time t is approximated by a continuous state stochastic process (diffusion process) {Xi(t) ] t 2 O}. For the fluctuation of jobs in a time interval we assume a normal distribution. In the steady-state, the density function fi(x) of the continuous state process can be shown to satisfy the Fokker-Planck equation [Koba74] assuming certain boundary conditions. A discretization of the density function gives the approximated product-form-like state probabilities ?i (Ici) for node i (see Fig. 10.2). K;(t)

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

+f(j&) A

Substitute the discrete process by a continuous diffusion process

Discretize the density function

t Xi@> +i (4 Determine the density function of the diffusion process for the steady state case Fig. 10.2

The principle

of the diffusion

approximation.

Although the derivation of this method is very complex, the method itself is very simple to apply to a given problem. Steady-state behavior of networks with generally distributed service and interarrival times at nodes can be approximately solved with the restriction of a single single server at each node. At present no solutions are available for multiple class networks [Mitz97]. Consider a GI/G/l queueing system with arrival rate X, service rate p, and the coefficients of variation cA and cg of the interarrival and service times, respectively. Using the diffusion approximation, the following approximated state probabilities can be obtained [Koba74, ReKo74]:

qk) =

l-P, { p(l - &P’,

k = 0, Ic > 0,

with:

(10.2)

424

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

and p = X/p < 1. Note that y is defined in Eq. (10.2) for convenience so that fi = exp (y ) . For the mean number of jobs we then have: (10.3) Differential equations underlying the diffusion approximation need boundary conditions for their solution. Different assumptions regarding these boundary conditions lead to different expressions. The preceding expressions are based on the work of [Koba74] and [ReKo74]. Using different boundary conditions, [Gele75] and [Mitz97] get different results that are more accurate than those of [Koba74] and [ReKo74] for larger values of utilization p. They [Gele75], [Mitz97] d erived the following expression for the approximated state probabilities: k = 0,

1 - P,

(10.4)

with ,6 and y as in Eq. (10.2). The mean number of jobs is then: K=p

pc; + c; 1+ 2u -4 1

1

Next we show how the diffusion approximation networks.

(10.5)

*

can be applied to queueing

10.1.1.1 Open Networks We can make use of the results for the GI/G/l system for each node in the network, provided that we can approximate the coefficient of variation of interarrival times at each node. We assume the following: l

The external arrival process can be any renewal process with arrival time 1/X and coefficient of variation CA.

inter-

l

The service times at node i can have any distribution time l/pi and coefficient of variation cgi.

l

All nodes in the network are single server with FCFS service strategy.

with mean service

According to [ReKo74] the diffusion approximation for the approximated state probabilities of the network have a product-form solution

it(kl, I I.

(10.6) i=l

NON-EXPONENTIAL

with the approximate marginal probabilities

DISTRIBUTIONS

425

as in Eq. (10.1):

jqk;) = l - Pi) 1pi(l- lj@-l,

I$= 0, k 2 1,

with X . ei

pi = -

Pi pi = exp

(10.8)

7

( -

w - Pi) c& * pi + c& > *

We approximate the squared coefficient of variation of interarrival node i using the following expression:

(10.9) times at

N

cf$ = 1+

cc j=o

C&

- 1) .p$. ej . e,l,

(10.10)

where we set: 2 -2

cBO -

(10.11)

cA*

For the mean number of jobs at node i we get: (10.12) k,=l

Similar results are presented in [Gele75], [GeMi80], and [Koba74]. Source

Sink

Fig. 10.3

A simple open queueing network.

Example 10.1 In this example (see Fig. 10.3), we show how to use the diffusion approximation for a simple open network with N = 2 stations. The

426

ALGORITHMS FOR NON-PRODUCT-FORM NETWORKS

external arrival process has mean interarrival time l/X = 2.0 and the squared 2 - 0 .94. The service times at the two stations have coefficient of variation cA the following parameters: pl = 1.1,

p2 = 1.2,

c;r = 0.5,

cg2 = 0.8.

With routing probabilities ~110= ~112= p21 = p22 = 0.5 and pal = 1, we compute the visit ratios, using Eq. (7.4): el

e2 = el . ~12 + e2 . p22 = 2.

= ~01 + e2 . ~21 = 2,

Node utilizations

are determined using Eq. (10.8): = 0.909,

X . e2 = 0.833. p2 = P2

We approximate the coefficients of variation of the interarrival the nodes using Eq. (10.10): 2 Gil = 1 + C(c& j=o

time at both

- l)p& * 3 el

= 1 + (ci - I)&

* z +
QPL

*z

+

(c”,, - l)p& * 2

= 0.920, 2 42 = 1 + c(c” Bj - 1)~;~ - 3 = 0.825. e2 j=o With Eq. (10.9) we get: & = exp

-

$2 = exp

-

at1- Pd c;1 * p1+ C&l > &

For the mean number of jobs zi,

K1=P1= 1 - bl

= 0.873,

21 - P2) = 0.799. - P2 + ($2 > we use Eq. (10.12) and obtain:

7.147,

K2 = A-S- = 4.151, 1 - p2

and the marginal probabilities are given by Eq. (10.7): %1(O)= 1 - pr = 0.091,

?r(k) = pr(1 - /?1)&-’ = 0.116.0.873k-1

%2(O)= 1 - p2 = 0.167,

for k > 0,

;r2(k) = p2(1 - @2)&1 = 0.167.0.799”-1

for k > 0.

NON-EXPONENTIAL

DISTRIBUTIONS

427

10.1.1.2 Closed Networks To apply the diffusion approximation to closed queueing networks with arbitrary service time distributions, we can use Eq. (10.7) to approximate marginal probabilities %i(ki), provided that the throughputs Xi and utilizations pi are known. In [ReKo74] the following two suggestions are made to estimate Xi:

1. For large values of K we use the bottleneck analysis: Search for the bottleneck node bott (node with the highest relative utilization), set its utilization to 1, and determine the overall throughput X of the network using Eq. (10.8). With this throughput and the visit ratios ei, compute the utilization of each node. The mean number of jobs is determined using Eq. (10.12). 2. If there is no bottleneck in the network and/or the number K of jobs in the network is small, replace the given network by a product-form one with the same service rates pi, routing probabilities pij, and the same K, and compute the utilizations pi. The approximated marginal probabilities given by Eq. (10.7) are then used to determine the approximated state probabilities: (10.13) where G is the normalizing constant of the network. Then for the marginal probabilities and the other performance measures improved values can be determined.

Fig. 10.4

A simple closed queueing network.

Example 10.2 Consider the closed network shown in Fig. 10.4 with the following parameters: p1 = 1.1,

p2 = 1.2,

c& = 0.5,

ci2 = 0.8,

and the routing probabilities: ~12

= 1 and

~21 = ~22 =

0.5.

For the visit ratios we use Eq. (7.5) and get: el = 1,

e2 = 2.

At first we choose K = 6 and use the bottleneck analysis to study the network.

428

ALGORITHMS FOR NON-PRODUCT-FORM

10.1.1.2.1

Solution with Bottleneck

NETWORKS

Let node bott be the bottleneck

Analysis

with the highest relative utilization: pb&t = m:x {pi} = A. m”;” ,

= x . max{0.909,1.667},

which means that node 2 is the bottleneck and therefore its utilization to 1:

is set

p2 = 1. Using Eq. (10.8) we determine the overall throughput

x = p2 *E For the utilization

1 Z-----E 1.667

X of the network:

0.6.

of node 1 we get:

pl=+

0.545.

Now we use Eq. (10.10) to determine the coefficients of variation of the interarrival times: 2

cam= 1 -t- (&I - l)pfl * el . ell + (c”,, - 1)&r . e2 . ell = 0.9, &2

= 1 + (431 - l)Pf2

* el * e;l

+ (~$3,

- l)pz, se2 . e;l = 0.7.

The pi are given by Eq. (10.9): fi1 = exp

( -

w &'Pl+c;,

- Pl)

= 0.3995,

&j = exp

>

(

W-P4 42.P2

Now, by using Eq. (10.12) we can compute the approximate mean number of jobs: = 0.908,

x2 = K - El

+&2

>

=

1.

for the

= 5.092.

Table 10.1 compares the values obtained by the preceding diffusion approximation (DA) with those obtained via DES. We can see that in this case the results match very well. Such close matching does not always occur, especially when one node in the network is highly utilized and the coefficients of variation are very small. To show the second method, we assume that there are K = 3 jobs in the network.

NON-EXPONENTIAL

Table 10.1

Node 1 Node 2

DISTRIBUTIONS

429

Comparison of results for the example (K = 6) Pz (DES)

z,

0.544

0.965 5.036

0.996

(DES)

Pi (DA)

Et

0.545

0.908

(DA)

1

5.092

10.1.1.2.2 Solution Using Product-Form Approximation Now we substitute the given network by a product-form one. To determine the utilizations of this network we can use the MVA and get:

pr = 0.501

and

p2 = 0.919.

The coefficients of variation for the interarrival the previous met hod: &

= 0.9

and

times remain the same as in

ci2 = 0.7.

For the ,!I&we use Eq. (10.9) and obtain: j& = exp

21

cil

-

Pl)

* p1+

= 0.35,

>

c&

p2

=

0.894.

The approximated marginal probabilities can be computed using Eq. (10.7): %1(O)= 1 - pr = 0.499, *r(l) = pr(l - ,&) = 0.326, fh(2)

= pr(1 - /%)br = 0.114,

?I (3) = p1(1 - /qg

= 0.039,

h(O)

= 1 - p2 = 0.081,

%(l)

= pz(1 - @a)= 0.0974,

+2(2) = ,oa(l - fi2)&

= 0.0871,

h(3)

= 0.0779.

= ,%(1 - &)p;

Now the following network states are possible: (3,0),

whose probability

(24,

(1,2),

(0,3),

is computed using Eq. (10.13): k(3,O)

= fq3)

Gf2(0).

?(2,1)

= 0.01111~ $

;r(l,2)

= 0.02839.

$,

+(0,3)

= 0.03887.

$

;

= 0.00316 - $

With the normalizing condition: fi(3,O) + ?(2,1)

+ q1,q

+ q&3)

= 1,

ALGORITHMS FOR NON-PRODUCT-FORM

430

NETWORKS

we determine the normalizing constant G: G = 0.00316 + 0.01111 + 0.02839 + 0.03887 = 0.08153, which is used to compute the final values of the state probabilities: ii(3,O) = 0.039,

+(2,1) = 0.136,

ii(l,2)

= 0.348,

qo, 3) = 0.477,

and with these state probabilities the improved values for the marginal probabilities are immediately derived: ~~(3) = ~~(0) = 0.039,

ni(2) = ~~(1) = 0.136,

q(1) = 7r2(2) = 0.348,

7rl(O) = 7r2(3) = 0.477.

For the mean number of jobs we get -

3

Ki = c

Ice TV

= 0.737

and

?7, = 2.263.

k=l

In Table 10.2 DES values for this example are compared to values from the preceding approximation. Table 10.2

Node 1 Node 2

Comparison of results for the example (K = 3). ~z (DES)

K, (DES)

pi (DA)

‘is7i (DA)

0.519 0.944

0.773 2.229

0.501 0.919

0.737 2.263

A detailed investigation of the accuracy of the diffusion method can be found in [ReKo74]. The higher the utilization of the nodes and the closer the coefficient of variation is to 1, the more accurate are the results. 10.1.2

Maximum

Entropy

Method

An iterative method for the approximate analysis of open and closed queueing networks is based on the principle of the maximum entropy. The term entropy comes from information theory and is a measure of the uncertainty in the predictability of an event. To explain the principle of the maximum entropy, we consider a simple system that can take on a set of discrete states S. The probabilities 7r(S) for different states are unknown; the only information about the probability vector is the number of side conditions given as mean values of suitable functions. Because in general the number of side conditions is smaller than the number of possible states, in most cases there is an infinite number of probability vectors that satisfy the given side conditions. The question now is

NON-EXPONENTIAL

DISTRIBUTIONS

431

which of these probability vectors shall be chosen as the one best suited for the information given by the side conditions and which is least prejudiced against the missing information. In this case the principle of maximum entropy says that the best-suited probability vector is the one with the largest entropy. To solve a queueing network using the maximum entropy method (MEM) the steady-state probabilities r(S) are determined so that the entropy function

H(T) = -C7r(S)lnn(S)

(10.14)

S

is maximized subject to given side conditions. The normalizing condition -the sum of all steady-state probabilities is l-provides another constraint. In [KGTA88], op en and closed networks with one or several job classes and -/G/l, -/G/m nodes are analyzed using this approach. Queueing disciplines such as FCFS, LCFS, or PS as well as priorities are allowed. An extension to the case of multiple server nodes is given in [KoA188]. For the sake of simplicity we only consider single class networks with -/G/l-FCFS nodes based on the treatment in [Kouv85] and [Wals85]. 10.1.2.1 Open Networks If we maximize the entropy by considering the side conditions for the mean number of jobs I?,, Eq. (7.18), and the utilization pi, Eq. (7.26)) then the following product-form approximation for the steady-state probabilities of open networks can be derived [Kouv85]: (10.15) where the marginal probabilities

are given by:

(10.16)

and where: (10.17) b, = c-i - pi z Ei ’ Gi zz e-i--

1 - pi *

(10.18) (10.19)

To utilize the MEM, we thus need the utilizations pi and the mean number of jobs Ki at each node. The equation pi = Xi/pi can be used to easily determine pi from the given network parameters. The mean number of jobs

432

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

-

Ki can be approximated as a function of the utilization pi and the squared coefficient of variation of the service and interarrival times [Kouv85]: K,

2

= pi

I+

c$i + Pic%i

l-pi

2

(10.20)

> ’

if the- condition (1 - c2Ai)/(l + c’,i) 5 pi < 1 is fulfilled. The computation of the Ki is made possible by approximating the squared coefficient of variation c~i of interarrival times. The following iterative expressions for computing the squared coefficients of variation is provided by [Kouv85] : (-ii = -1+

C3i

(10.21)

($.gql~

= 1 + pji (Cgj - 1) )

c$i = pi(l

(10.22)

- pi) + (1 - pi)Cii

There are other approximations of jobs ITi and for the estimation variation. For more information on With this information, the MEM can be summarized in the following

(10.23)

+ pfC:i*

for the computation of the mean number of the necessary squared coefficients of these equations, see Section 10.1.3. for the analysis of open queueing networks three steps:

Determine the arrival rates Xi using the traffic equations and the utilizations pi = Xi/pi for all nodes i = 1, . . . , N. Determine -1 arrival

the squared coefficients

(Eq. (7.1))

of variation.

Initialize. The squared coefficients of variations times are initially set to one for i = 1,. . . , N.

c2Ai of the inter-

1 STEP 2.2 ] Compute the squared coefficients of variation c2oi of the interdeparture times of node i for i = 1, . . . , N using Eq. (10.23). Compute the squared coefficients of variation c& of node i for i = 1, . . . , N and j = 0, . . . , N using Eq. (10.22). From these the new values of the squared coefficients of variation cii of the interarrival times are computed using Eq. (10.21). 1 STEP 2.3 ] Check the halting condition. If the old and new values for the c2Ai differ by more than E, then go back to Step 2.2 with the new & values. Determine the performance measures of the network beginning with the mean number of jobs ??i for i = 1,. . . , N using Eq. (10.20) and then compute the maximum entropy solutions of the steady-state probabilities using Eq. (10.15). The MEM normally approximates the coefficients of variation less accurately than the method of Kiihn (see Section 10.1.3) and therefore the method of Kuhn is generally preferred for open networks with a single job class. The MEM is mainly used for closed networks.

NON-EXPONENTIAL

DISTRIBUTIONS

433

IO. 1.2.2 Closed Networks For closed networks that consist only of -/G/lFCFS nodes, the following product-form formulation for the approximated steady-state probabilities can be given if we maximize the entropy under the side conditions on the mean number of jobs ??i and the utilization pi:

7r(kl. . .$N) = A- Aqkl). . . . - cv@N), G(K)

(10.24)

with: (10.25) and:

G(K) =

F&)

c

. . . . . FN(kN).

(10.26)

As in the case of open networks, coefficients ai and bi are respectively given by Eqs. (10.17) and (10.18). Th e computation of the steady-state probabilities of closed networks is more difficult than that for open networks as the pi and 17, values of the individual nodes cannot be determined directly. Therefore we use the following trick: We construct a pseudo open network with the same number of nodes and servers as well as identical service time distribution and routing probabilities as the original closed network. The external arrival rates of this open network are determined so that resulting average number of jobs in the pseudo open network equals the given number of jobs K in the closed network: N c

11; = K,

(10.27)

i=l

where x: denotes the mean number of jobs at the ith node in the pseudo open network. The pseudo open network arrival rate X is determined iteratively by using Eqs. (10.20) and (10.27), assuming that the stability condition Xei max
434

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

Eq. (10.25), th e normalizing constants as well as the utilizations and mean number of jobs at each node of the closed network can be computed. This approximation uses values for the coefficients ai and bi, which are computed for the pseudo open network. To compensate the error in the approximation, we apply the work rate theorem (Eq. (7.7)), which gives us an iterative method for the computation of the coefficients ai : .(n+l) z

=

a,!“’

. K

(10.28)

. (pi.p$

. p;

with the initial values: (10.29)

For the coefficients bi we have: (10.30) The asterisk denotes the performance measures of the pseudo open network. The algorithm can now be given in the following six steps. Compute the visit ratios ei for i = 1,. . . , N, using Eq. (7.2) and the relative utilizations xi = ei/pi. Construct and solve the pseudo open network. r-1 Initialize. Initial value for the squared coefficients of the interarrival times at node i = 1, . . . , N:

Initial value for the arrival rate of the pseudo open network: x =

Oagg lCbott ’

where x&t

= max {xi} is the relative utilization i

of the bottleneck node.

[I Compute X from Condition (10.27) by using Eq. (10.20). Thus, X can be determined as solution of the equation: 1 +

Csi

+

X * Xi

* C~i

-K=O,

(10.31)

I.-X-Xi

if (1 - C~i)/(l + c”,i) 5 X . x; < 1 for all i is fulfilled. To solve the non-linear equation, we use the Newton-Raphson method.

NON-EXPONENTIAL

DISTRIBUTIONS

435

-1 Compute for i, j = 1,. . . , N the squared coefficients of variation 2 using the Eqs. (10.21)-(10.23). For this computation we need c&, C$ and & the values pi = X-Q and Xi = X.ei. p-1 Check the halting condition. If the new values for the & differ less than E from the old values, substitute them by the new values and return to Step 2.2. m% Determine for i = 1, . . . , N the utilizations pi = (A . ei)/y;, and with Eq. (10.20) the mean number of jobs l?r in the pseudo open network. Determine the coefficients ai and b; of the pseudo open network using Eqs. (10.29) and (10.30). w

Solve the closed network.

-1 Use the convolution method to determine the normalizing constant G(K) of the network. The functions Fi(ki) are given by Eq. (10.25). -1 Determine the performance parameters pi and ??i of the closed network. For this we use the normalizing constant Gc’ (Ic) as defined in Eq. (8.8). From Eq. (8.7) we have ni(k) = Fi(lc) +G$$(K - Ic)/G(k) and therefore: (10.32) (10.33)

(7 Use Eq. (10.28) to determine a new value for the coefficients ai. If there is a big difference between the old values and the new values, then return to Step 5.1. -1 Use Eq. (10.24) to determine the maximum entropy solution for the steady-state probabilities and other performance parameters of the network ils, e.g., Xi = /hipi or Ti = Xi/X,.

Fig. 10.5

Closed network.

The maximum entropy method is now illustrated on a simExample 10.3 ple network with N = 2 nodes and K = 3 jobs (see Fig. 10.5). The queueing

436

ALGORITHMS FOR NON-PRODUCT-FORM NETWORKS

discipline at all nodes is FCFS and the service rates and squared coefficients of variation are given as follows: 2 cB2 = 0.5.

2

and

p2 = 2,

Pl = 1,

cl?1 = 5,

The analysis of the network is carried out in the following steps. As condition for the termination of iterations we choose E = 0.001. Determine the visit ratios and relative utilizations: and

el = e2 = 1,

X2=05Aa

Xl =I,

Construct and solve the pseudo open network. I STEP 2 I

Initialize: x = 0.99/x1 = 0.99.

c& = ci2 = 1,

-1 Determine X. We use the Newton-Raphson iteration method to solve the following equation:

2 -x-xi 1+c&+x*xi*c& -3=o 1-X.X; > z 2 ( and get X = 0.5558. -1

Determine the squared coefficients of variation: &

= pl(l - ~1) + (1 - pl)c;,

3j2 = p2(1 - ~2) + (I2

+ p~c& = 2.236,

,a)~;,

+ p;~;~

= 0.961,

cl2 = 1 + ~12 s (c”,, - 1) = 2.236, 2 c21 = 1 + pzl . (4,

&

=

-1 +

ci2 = -1+

- 1) = 0.961,

x2 ’ P21

(

k

Xl * P12 x2 * (42

(-2.4

-l

= o g61

--I

= 2 236

(%, + 1) >

+ 1) >

LT

A.

Check the halting condition: 1Cy;f-) - c?;ld) 1= (0.961 - l( = 0.039 > E, Icy - c~;~~)I = 12.236- 11= 1.236 > E.

As the halting condition is not fulfilled, we return to Step 2.2 and determine a new value for the constant X.

NON-EXPONENTIAL

DISTRIBUTIONS

437

After 14 iterations, the halting condition is fulfilled and we get the following results: 2 CA1 = 2.130,

x = 0.490,

ci2 = 2.538.

Determine pi* and xz:

= 2.446, Determine the coefficients: ~ Pi . ~ PT = 0.241, a1 = 1 - pT KT - pi

- Pa . ~ PZ = 0.257, a2 = 1 - p; 17; - p;

b =K;-d -

bz =

1

-*

=‘)

o8

Xi-p; - E*

= La 0 558 2

Kl

Solve the closed network. -1 Determine the normalizing constant using the convolution method (see Section 8.1). For the functions Fi(Ici), i = 1,2, we use Eq. (10.25) and get: wo

=

F2(0)

I,

= L,

I-$(l) = 0.193, Fr(2) = 0.154,

Fz(1) = 0.144,

&(3)

F2(3) = 0.045.

= 0.123,

F2(2) = 0.080,

The procedure to determine the G(K) is similar to the one in Table 8.2 and summarized in the following tabular form: k

2

\n 0

1 0.193 0.154 0.123

1 2 3

1 0.3362 0.2618 0.2054

For the normalizing constant we get G(K) = 0.2054. -1 Determine pi and xi. For this we need the normalizing constant G!‘(5) of the network with Node i short-circuited. We have only two nodes and therefore it follows immediately that: G(l% nr

= Fz(k),

G% iv

= Fl (k) 3

438

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

and, therefore: p1 = 1 - @ Fl@)

-k=l

p2 z 1 - @#

= 0.782,

= L, 0 400

(1)

. G, (K - k) = 2.090,

G(K)

- k) = :* 0 910 [I

Determine the new coefficients ai. With Eq. (10.28) we get:

%

(new) _-. al +KS p; . (&$

=0.242,

(new) _-a2+K.p;+

=o.253.

a2

(p2.$$’

Because the old and new values of the ai differ more than E = 0.001, the analysis of the closed network has to be started again, beginning with Step 5.1.

After altogether three iteration steps the halting condition is fulfilled and we get the following final results: p1 = 0.787,

p2 = 0.394,

and

x1 = 2.105,

K2 = 0.895.

Compute all other performance measures: X1 = /xlpl = 0.787, T1 = K1/X1 = 2.675,

X2 = p2p2 = 0.787, T2 = J&/x:!

= 1.137.

To compare the results, we want to give the exact values for the mean number of jobs: x1 = 2.206,

K2 = 0.794.

The method of Marie (see Section 10.1.4.2) gives the following results: x1 = 2.200,

K2 = 0.800.

The closer the service time distribution is to an exponential distribution (c& = 1) , the more accurate are the results of the MEM. For exponentially distributed service and arrival times the results are always exact. Another important characteristic of the MEM is the fact that the computation time is relatively independent of the number of jobs in the network.

NON-EXPONENTIAL

10.1.3

Decomposition

for Open

439

DISTRIBUTIONS

Networks

This section deals with approximate performance analysis of open non-product-form queueing networks, based on the method of decomposition. The first step is the calculation of the arrival rates and the coefficients of variation of the interarrival times for each node. In the second step, performance measures such as the mean queue length and the mean waiting time are calculated using the GI/G/l and GI/G/ m f ormulae from Sections 6.12 and 6.14. The methods introduced here are those due to Kuhn [Kiihn’lS, Bolc89], Chylla [Chy186], Pujolle [PuAi86], Whitt [Whit83a, Whit83b] and Gelenbe [GePu87]. Open networks to be analyzed by these techniques must have the following properties: distributed

and

l

The interarrival times and service times are arbitrarily given by the first and second moments.

l

The queueing discipline is FCFS and there is no restriction on the length of the queue.

l

The network can have several classes of customers (exception is the method of Kuhn).

l

The nodes of the network can be of single or multiple server type.

l

Class switching is not allowed.

With these prerequisites the method works as follows: Calculate the arrival rates and the utilizations of the individual nodes using Eqs. (10.35) and (10.36), and (10.37) and (10.38), respectively. Compute the coefficient of variation of the interarrival each node, using Eqs. (10.41)-( 10.44).

times at

Compute the mean queue length and other performance measures. We need the following fundamental formulae (Step 1); most of them are from Section 7.1.2: Arrival rate Xij,T from node i to node j of class r:

Arrival rate Xi,, to node i of class r: N

k,r

= XOi,r + C j=l

Xj,r * Pji,r-

(10.35)

440

ALGORITHMS FOR NON-PRODUCT-FORM NETWORKS

Arrival rate Xi to node i: (10.36) Utilization

pi,r of node i due to customers of class T: pi,r = ~. mi f Pi,r

Utilization

(10.37)

pi of node i: R

Pi = C r=l

(10.38)

Pi,r*

The mean service rate pi of node i: 1

(10.39)

Coefficient of variation cgi of the service time of node i: 2

&

= -1+

R

Air A. r=l Ai c

(

&

>

(10.40)

* @fJ- + l),

with coefficient of variation ci,r for the service time of customers of class r at node i. The calculation of the coeficient of variation or interarrival times (Step 2) is done iteratively using the following three phases (see Fig. 10.6): 0 1291) C2z,l> \

. . .

VNi,R>

CK,,,)

(hl,l, (Lc;z)

)

(X2)z,CD,

. . .

-:

/

Phase 1: Merging

&,

(~N,R,

Phase 2: Flow

$~v,R)

Phase 3: Splitting

Fig. 10.6 Phases of the calculation of the coefficient of variation.

l

Phase 1: Merging Several arrival processes to each node are merged into a single arrival process. The arrival rate X is the sum of the arrival rates of the individual arrival processes. For the coefficient of variation of the interarrival time, the several authors suggest different approximate formulae.

NON-EXPONENTIAL

441

DISTRIBUTIONS

Phase 2: Flow

l

The coefficient of variation cg of the interdeparture times depends on the coefficients of variation of the interarrival times CA and the service times cg . Here again the different authors provide different approximate formulae. Phase 3: Splitting

l

For the splitting of the departure process, all authors apply the formula: = 1 +pij,r

cFj,r

* (c& - 1).

(10.41)

The iteration starts with phase 1 and the initial values are: cij,r = 1. The corresponding formulae for phases 1 and 2 are introduced in the following: l

Decomposition of Pujolle

[PuAi86]:

Merging:

*A j,r ’ Pji,r + &i,r ’

2 _ 1 R cAi - x i - cr=l

‘ii,r

XO,r

* POi,r

,

(10.42)

(10.43)

* k,r-

Flow: ‘Y&i

l

= Pf ’ <&i + l) + (l - Pi) ’

Decomposition of Whitt

C~i

+ Pi * (1 - 2pi).

(10.44)

[Whit83b, Whit83a]:

Merging: see Pujolle. Flow: 2 = 1 + ” ’ (~Cgi

l

‘) + (1 - pi) ’ (C~i - 1).

(10.45)

Decomposition of Gelenbe [GePu87]: Merging: see Pujolle. Flow:

c& = -1 + Ai 5 !Q * (c&

+ 1) + (1 - pi) * (Cii + 1+ 2 * Pi>.

r=l CLG~

(10.46)

442

ALGORITHMS FOR NON-PRODUCT-FORM NETWORKS l

Decomposition of Chylla

[Chyl86]:

N xj,r * Pji,r 2 x, CA+ = 1 + c 2,r j=o

* gi,,

- l>,

(10.47)

(10.48)

Flow: 2

cgi = 1 + P& (pi) +(& - 1) + (1 - P:,(pi))

fig. 10.7 l

Decomposition of Kuhn

Substitution

*
(10.49)

of a feedback.

[Kiihn79]:

Before starting with the iteration in the method of Kuhn, nodes with direct feedback are replaced by nodes without feedback (see Fig. 10.7 and Eq. (10.50)). x; = Xi(l - Pii), CL: = pip - Pii), Pij & = ~1 _ pii for j f i, p,’ = 0.

cg = pii + (1 - p&‘,,,

(10.50)

c& =

2&i

* x2i * (Ali +

A%)

* (I1 + I2 + 13 + I*) - 1.

(10.51)

Only two processes can be merged by this formula. If there are more processes to merge, then the formula must be used several times. The terms Zl,l = 1,2,3,4 are functions of the Xji and the cj;. For details see [Kuhn791 or [Bolc89].

NON-EXPONENTIAL

443

DISTRIBUTIONS

Flow: c& = Cii + a&&

- pf (&

+ &)

(10.52)

- GKLB.

For GKLB see Eqs. (6.72) and (6.92). To calculate the mean queue length (Step 3) and the other performance measures, GI/G/l and GI/G/ m f ormulae from Sections 6.12 and 6.14 can be used in the case of a single class of customers. Here we give the extension of the most important formulae to the case of multiple classes of customers: M/M/m-FCFS:

l

(10.53) with the probability

of waiting, P,;, given by Eq. (6.28)

Allen-Cunneen [AllegO] :

l

(10.54) l

Kramer/Langenbach-Belz

[KrLa76]:

Q~,TKLB

= Qi,rAC

(10.55)

’ GKLB,

with the correction factor: --.-.

G KLB

ei

2 3

(1

-

pi)

C1

Pi

Gi,rj2

(CL,, + C”,i,,>

c5i,r 5 ‘7

= (4,r -

i e l

-

(

(’

-

pi)

’ (C”,i,,

-

(10.56)

1)

+ C~i,~)

c;i,r > 1.

7

Kimura [Kimu85] :

<‘Zli,r-

. l-4,

T

l-4C(?&,)

*e

2 (l--Pi) Pi ( -3

)

+

4i,r) l-4,

+

T

1fChh

+ cf&

+ c&.

-

1’

(10.57) with the coefficient of cooperation C(mi,

pi) = (1 - pi)(mi

-l> J275&2 16mip;

’

(10.58)

444

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

Sink

Fig. 10.8 Open non-product-form queueing network. l

Marchal [Marc78]:

(1+ C”,i,,>*(c&l. + P%i,?-1+GKLB. Qi,rMAR z5Qi,rM,M/rn 20 + P5c;i.J

(10.59)

Example 10.4 Consider the open non-product-form queueing network of Fig. 10.8. The routing probabilities of the network are given by: ~13 = 0.5,

~12 = 0.5,

p31

=

0.6,

p21

=

p41 = 1.

Jobs are served at different nodes with rates: ~1 =

25,

p2 = 33.333

/!.L3 =

16.666,

20,

/A4 =

and arrive at node 4 with rate X04 = 4. The coefficients of variation of the service times at different nodes as well as of the interarrival times are given by:

c& =

2.0,

c& =

6.0,

c& =

0.5,

c&

= 0.2,

Co4 2 -

4.0.

With Eq. (10.35) we can determine the arrival rate at each node: Xl =2J,

x2 =a,

x3 =lo,

and using Eq. (10.37) we determine the utilizations p1=0.8,

p2=0.3,

p3 = 0.6,

x4 =4,

of different nodes: 0.2 p4 = -*

The initial values for the coefficients of variation cij are: Cf2 = CT3= c;1 = c;1 = c& = 1. Now we start the iteration using the decomposition of Pujolle to obtain the different coefficients of variation of the interarrival times c;~, i = 1,2,3,4.

NON-EXPONENTIAL

DISTRIBUTIONS

Iteration 1: Merging (Eq. (10.42)):

+ c41 X4P41) d, = &h?1)21 + & X3P31 = &lOl+llO~O.6+1~4l) YZ1. Similarly, we obtain:

Flow (Eq. (10.44)):

611= Pa3, + 1)+ (1- Pl)zl, + /a(1-

2Pl) = 0.82(2 + 1) + (1 - 0.8) +1 + 0.8 . (1 - 2 -0.8) =1.64.

Similarly, we get:

c& = 1.45, c& = 0.82, Splitting

c& = 3.368.

(Eq. (10.41)): CT2= 1+m2+201 -1) = 1 + 0.5 . (1.64 - 1) = 1.32, 2

cl3 = 1.32,

c& = 1.45,

c31 = 0.892,

c& = 3.368.

Iteration 2: Merging (Eq. (10.42)): cil = 1.666,

c;2 = 1.32,

ci3 = 1.32,

ci4 = 4.0.

Flow (Eq. (10.44)): cg, = 1.773, Splitting

c& = 1.674,

c& = 0.948,

c& = 3.368.

(Eq. (10.41)):

2 cl2 = 1.387,

cf3 = 1.387,

c& = 1.674,

c& = 0.969,

c& = 3.368.

After six iteration steps we get the values for the coefficients of variation of the interarrival times ci, at the nodes, shown in Table 10.3. In Table 10.4 the mean number of jobs at different nodes are given. We use the input parameters

446

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

Table 10.3 Coefficients of variation cii, i = 1,2,3,4 of the interarrival times at the nodes Iteration 1 2 3 4 5 6

1.0 1.666 1.801 1.829 1.835 1.836

4,

4,

4,

1.0 1.320 1.387 1.400 1.403 1.404

1.0 1.320 1.387 1.400 1.403 1.404

4.0 4.0 4.0 4.0 4.0 4.0

Table 10.4 Mean number of jobs ??, at node i using the formulae of Allen-Cunneen (AC), KrgmerLangenbach/Belz (KLB) , and discrete-event simulation (DES) Methods

EI

772

773

774

AC KLB DES

6.94 5.97 4.36

0.78 0.77 0.58

1.46 1.42 1.42

0.31 0.27 0.23

and values for the coefficients of variation for the interarrival times at the nodes after six iterations (given in Table 10.3). As we can see in Table 10.4, we get better results if we use the Kramer-Langenbach/Belz formula instead of the Allen-Cunneen formula. In this second example we have two servers at node 1. The Example 10.5 queueing network model is shown in Fig. 10.9. Source z* << < ”*:” c7

Sink Fig. 10.9

Open non-product-form

queueing network with two servers at node 1.

The service rate of each server at node 1 is ,~i = 12.5. All other input parameters are the same as in Example 10.4. We also have the same values for the arrival rates Xi and the utilizations pi. The initial values for the

NON-EXPONENTIAL

DISTRIBUTIONS

447

coefficients of variation c;j are again: 2 Cl2

=

2 Cl3

=

2 c21

= c& = CT;1= 1.0.

This time we use the formula of Whitt to determine the coefficients of variation cAi iteratively. Iteration 1: Merging: Whitt uses the same merging formula as Pujolle (Eq. (10.42)) and we get:

Flow (Eq. (10.45)): c& = l+

P?Gl - 1)+

(1 - Pxi, - 1) fi = 1 + O-64(2 - 1) + (1 - 0.64)(1 - 1) l.h = 1.453

c& = 1,45, Splitting

cg3 = 0.82,

c& = 3.848.

(Eq. (10.41)):

cT2 = 1.226,

cT3 = 1.226,

c;i = 1.450,

c& = 0.892,

& = 3.848.

Iteration 2: Merging (Eq. ( 10.42)) : cil = 1.762,

cl2 = 1.226,

c;, = 1.226,

c;, = 4.0.

Flow (Eq. (10.45)):

CL1= 1.727, Splitting

c& = 1.656,

c& = 0.965,

c& = 3.848.

(Eq. (10.41)):

2 cl2 = 1.363,

cT3 = 1.363,

c;i = 1.656,

c& = 0.979,

c& = 3.848.

After seven iterations, we get the values for the coefficients of variation of the interarrival times ci, at the nodes shown in Table 10.5. In Table 10.6, the mean number of jobs at the different nodes are given. We use the input parameters and values for the coefficients of variation for the interarrival times at the nodes after seven iterations (given in Table 10.5). These results are similar to the results of Example 10.4, and the differences with discrete-event simulation results are slightly smaller than in Example 10.4.

448

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

Table 10.5 Coefficients of variation cii of the interarrival at the nodes

4,

Iteration 1 2 3 4 5 6 7

times

1.0 1.762 1.891 1.969 1.983 1.991 1.992

1.0 1.226 1.363 1.387 1.401 1.403 1.405

4.0 4.0 4.0 4.0 4.0 4.0 4.0

1.0 1.226 1.363 1.387 1.401 1.403 1.405

Tab/e 10.6 Mean number of jobs ??, at the nodes using the formula of Allen-Cunneen (AC), Krtimer-Langenbach/Belz (KLB), and discrete-event simulation (DES)

AC KLB DES

10.1.4

Methods

6.48 6.21 4.62

for Closed

0.78 0.77 0.57

1.46 1.42 1.38

0.31 0.27 0.23

Networks

10.1.4.1 Robustness for Closed Networks In case of closed non-product-form queueing networks with -/G/l and -/G/m FCFS nodes, the easiest way to analyze them is to replace the -/G/l and -/G/m FCFS nodes by -/M/l and -/M/m FCFS nodes and to use a product-form method such as MVA, convolution, or the SCAT algorithm. This substitution can be done due to the property of robustness of such closed non-product-form queueing networks. Robustness, in general, means that a major change in system parameters generates only minor changes in the calculated performance measures. In our case it means that if we replace the values of the coefficients of variation of the service times c& (i = 1,. . . , N) by 1, i.e., we assume exponentially distributed service times instead of arbitrarily distributed service times, we obtain a tolerable deviation in the calculated performance measures. We have investigated more than 100 very different closed non-productform queueing networks with -/G/l and -/G/m FCFS nodes. The maximum number of nodes was 10 and the coefficient of variation varied from 0.1 to 15. For a network with only -/G/l FCFS nodes, the mean deviation is about 6% for both single and multiple class networks. In the case of only -/G/m FCFS nodes, the deviation decreases to about 2%. If we have only -/G/co nodes, the deviation is zero. The difference between single and multiple class networks is that the maximum deviation is greater for multiple class networks (40% instead of 20% for single class networks).

NON-EXPONENTIAL

DISTRIBUTIONS

449

For open non-product-form queueing networks, the influence of the coefficients of variation is much greater, especially if the network is a tandem one without any feedback. So we cannot suggest to use the property of robustness to obtain approximate results for open non-product-form queueing networks. It is better to use one of the approximation methods introduced in Section 10.1.2. Example 10.6 In this example we study the robustness property of several closed networks with different coefficients of variation. Consider the three network topologies given in Fig. 10.10 together with four combinations of the coefficients of variation (see Table 10.7). Note that the first combination (a) corresponds the case of all service times being exponentially distributed and the corresponding network being a product-form one. The analysis of the resulting models is done with K = 5 and K = 10 jobs in the system. Table 10.7

Squared coefficients of variation for the net-

works in Fig. 10.10 Combinations

Node 1

Node 2

Node 3

41

42

43,

z

0.2 1.0

0.4 1.0

0.8 1.0

1

4.0 2.0

4.0 1.0

8.0 0.2

We use two combinations of service rates. The first combination leads to a more balanced network, while the second combination results in a network with a bottleneck at node 1 (see Table 10.8). Tab/e 10.8

Two combinations of service rates

Balanced network Network with bottleneck

0.5 0.5

0.333 1.0

0.666 2.0

In Tables 10.9 and 10.10, the throughputs X for the three network topologies together with the different combinations of the coefficient of variation are given. If we compare the product-form approximation results (case a, Table 10.7) with the results for non-product-form networks (cases b, c, and &, Table 10.7), given in Tables 10.9 and 10.10, we see that the corresponding product-form network is a good approximation to the non-product-form one provided: l

The coefficients of variation are < 1 4.

(case

b) and/or not too big (case

450

ALGORITHMS FOR NON-PRODUCT-FORM

Network

NETWORKS

1

--l-/-F

)

p12 =0.6;

p13 =0.4

r Network 2

~12

=

P21 =

0.6; p13 = 0.4 0.3;

p23

= 0.7

Network 3

Pll

Fig. 10.10

= 0.2; p12 = 0.4;

p13

= 0.4;

P21 = 0.3;

P22 = 0.2; p23 = 0.5;

p31 = 0.6;

p32 = 0.2;

Different closed networks.

~3~

= 0.2;

NON-EXPONENTIAL

Table 10.9

Throughputs

of Variations

451

X for the networks in Fig. 10.10, for the balanced case Network

Squared Coefficient

DISTRIBUTIONS

K=5

1

Network 3

Network 2

K = 10

K=5

K = 10

K=5

K = 10

E

0.47 0.43

0.50 0.47

0.45 0.41

0.49 0.47

0.40 0.37

0.44 0.42

:

0.37 0.39

0.44 0.42

0.33 0.48

0.39 0.47

0.29 0.36

0.34 0.41

Tab/e 10.10

Throughputs

X for the networks in Fig. 10.10, for the case with a bottle-

neck. Squared Coefficient of Variations

Network K=5

1

Network 3

Network 2

K = 10

K=5

K = 10

K=5

K = 10

;

0.50

0.50

0.52 0.51

0.50

0.50

0.52

zi

0.49 0.47

0.50

0.47 0.52

0.50

0.48 0.49

0.50 0.52

l

The network has a bottleneck.

l

The number of jobs in the network is high.

The larger the coefficients of variation (c& > I), the larger are the deviations. Example 10.7 In this example we consider in more detail the dependency of the approximation accuracy on the number of jobs K in the network. The examination is based on the first network given in Fig. 10.10 together with the same input parameters except that ,~i = 1, for i = 1,2,3. The results for the throughputs X for different numbers of jobs K are shown in Table 10.11. As we can see in Table 10.11, the larger the number of jobs K, the better is the approximation. Table 10.11 Throughput X for the Network 1 shown in Fig. 10.10, for different values of K and combinations of the cii K

3

4

5

10

20

50

E

0.842 -

0.907 0.970

0.940 0.991

0.996 1.00

1.00

1.00

zi

0.716 -

0.856 0.766

0.894 0.805

0.972 0.917

0.998 0.984

1.00

452

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

10.1.4.2 Marie’s Method If the network contains single or multiple server FCFS nodes with generally distributed service times, then an approximate iterative method due to Marie [Mari79, Mari80] can be used. In this method each node of the network is considered in isolation with a Poisson arrival stream having load-dependent arrival rates Xi (Ic). To determine the unknown arrival rates Xi (Ic), we consider a product-form network corresponding to the given non-product-form network. The product-form network is derived from the original network by simply substituting the FCFS nodes with generally distributed service times by FCFS nodes with exponentially distributed service times (and load-dependent service rates). Because the service rates pi(lc) for the nodes of the substitute network are determined in the isolated analysis of each node, the method is iterative where the substitute network is initialized with the service rates of the original network. The load-dependent arrival rates Xi(k) of the nodes 1, , . . , N can be computed by short-circuiting node i in the substitute network (see Fig. 10.11). All other nodes of the network are combined into the composite node c (see Section 8.4).

fig. 10.11 Short-circuiting node i in the substitute network. If there are k jobs at node i, then there are exactly K - k jobs at the composite node c. The load-dependent service rate through the substitute network with K - k jobs and short-circuited node i is, therefore, the arrival rate at node i with k jobs at node i, which is: Xi(k)

= Ap)(K

- k),

for k = 0,. . . , (K - 1).

(10.60)

Here Xp)(K - k) denotes the throughput of the network with node i shortcircuited. This throughput can be computed using any algorithm for a product-form network. The arrival rates X;(k) can now be used for the isolated analysis of the single node of the network. The state probabilities ni(k) of the nodes in the substitute network and the values of the load-dependent service rates p;(k) for the next iteration step have to be computed. It is shown in [Mast771 that in the considered network for each isolated node, a partition of the state space can be found so that the following condition holds: vi(k)

. xi(k)

= Xi(k - 1) . ni(k - 1).

(10.61)

According to this equation the probability that a job leaves a node in state k is equal to the probability that a job arrives at the same node when this node is in state k - 1. The stochastic process, which can be assigned to the node

NON-EXPONENTIAL

DISTRIBUTIONS

453

under consideration, behaves exactly like a birth-death process with birth rates Ai and death rate vi(lc) (see Section 3.1). The state probabilities, which we are looking for, can therefore be computed as follows: k= l,...,K,

(10.62) (10.63)

The new load-dependent service rates for the nodes in the substitute network for the next iteration step are then chosen as follows: pi(k) = vi(k)

for i = l,...,N.

(10.64)

Now the main problem is to determine the rates vi(lc) for the individual nodes in the network. The fact that in closed queueing networks the number of jobs is constant allows us to consider the individual nodes in the network as elementary X(lc)/G/m/K st at ions in the isolated analysis. The notation describes a more specific case of a GI/G/ m node where the arrival process is Poisson with state-dependent arrival rates and for which an additional number E( exists, so that X(lc) > 0 for all Ic < K and X(lc) = 0 for all Ic 2 K. If we do the isolated analysis, we have to differentiate between the following l

Product-form

node

l

Single server node with FCFS service strategy and generally distributed service time

In the first case, a comparison of the Eqs. (10.62) and (10.63) with the corresponding equations to compute the state probabilities of product-form nodes with load-dependent arrival rates [Lave83] shows that the rates vi(lc) correspond to the service rates pi(k). For nodes with exponentially distributed service times and service discipline PS, IS, LCFS PR, respectively, it follows immediately from Eq. (10.64), that the service rates of the substitute network remain unchanged in each iteration step. Next, consider FCFS nodes with generally distributed service times, which can be analyzed as elementary X(k)/G/l/K-FCFS systems. Under the assumption that the distribution of the service time has a rational Laplace transform, we use a Cox distribution to approximate the general service times. Depending on the value of the squared coefficient of variation c‘& of the service time, different Cox models are used: if c& = l/m&& for m = 3,4, . . ., where E is a suitable tolerance area, then we use an Erlang model with m exponential phases, If ~5; >_ 0,5, then a Cox-2 model is chosen. For all other cases a special Cox model with m phases is chosen.

ALGORITHMS FOR NON-PRODUCT-FORM

454

NETWORKS

According to [Mari80], the parameters for the models are determined as follows: l

For a Coz-2 model (Fig. 10.12):

Fig. IO.12

l

Cox-2 model.

The special Cox-m model is shown in Fig. 10.13. A job leaves this model with probability bi after completing service at the first phase or runs through the rest of the (m - 1) exponential phases before it is completed. The number m of exponential phases is given by:

11 1 7j-.

m=

CBi

”

,

Fig. IO.13

”

**

Special Cox-m model.

From Section 1.2.1, we have: b, = 2mc& a /%

=

+ (m - 2) - j/m2

2(m -

For the Erlang-m

+ 4 - 4mcgi

+ 1)

9

[m - bi(m - l)] /%i*

Here bi denotes the probability phase. l

1)(c2Bi

that a job leaves node i after the first

model we have:

NON-EXPONENTIAL

DISTRIBUTIONS

455

General Cox-m model.

Fig. 10.14

Next consider the general Cox-m model, shown in Fig. 10.14. Each of the elementary X(k)/C,/l/K no d es can be assigned a CTMC whose states (k, j) indicate that the service process is in the jth phase and k jobs are at node i with corresponding steady-state probability ~(k, j). The underlying CTMC is shown in Fig. 10.15. The corresponding balance equations for each node i are given by: (10.65) j=l

x&k - 1)7ri(lc - 1,l) + &,&,j?ri(k

+ l,j)

j=l =

Xi(lc)%(b

1)

+

i&,1+,

x&k - 1)7r& - 1,j) + a;$-i&j-i7ri(k,j = h(+G(k?>

11,

(10.66)

- 1)

+ fii,jT(kj).

(10.67)

Let xi(k) denote the probability that there are k jobs at node i and vi(k) the conditional throughput of node i with k jobs: (10.68) j=l F Vi(k)

=

j=l

h,jl%j%

(k

j>

m

Jg 746 8

(10.69) ’

Here b,,j denotes the probability that a job at node i leaves the node after the jth phase. If we add Eq. (10.66) to Eq. (10.67) (for j = 2,. . . , m), we obtain for node i:

j=l

j=l

m

m

456

ALGORITHMS FOR NON-PRODUCT-FORM

Fig. 10.15

NETWORKS

CTMC for the general Cox-m model.

NON-EXPONENTIAL

457

DISTRIBUTIONS

Now we can apply Eq. (10.69) to Eq. (10.70) and obtain: &(k-l).~s(k-l,j)+u,(k+l).~?r~(k+l,j) j=l

j=l

= b(k)

* &i(k’j) j=l

+ Q‘q

&i(k, j=l

j),

Xi(k - l);lri(k - 1) + Vi(k + 1)7ri(k + 1) = Xi(lc)7ri(k) + Vi(k)7l’i(k). If we now consider Eq. (10.65), we finally get for the solution of an elementary X(k)/C,/l/K system in steady-state [Mari78]: Ui(k)7ri(k) = Xi(k - 1)7ri(k - 1). But this is exactly Eq. (10.61). For this reason the state probabilities of an FCFS node with generally distributed service time can be determined using Eqs. (10.62) and (10.63) w h ere Eq. (10.69) is used to compute the conditional throughputs vi (k) of the single nodes. To determine the conditional throughputs, efficient algorithms are given in [Mari80]. In the case of an exponential distribution, vk(k) is just given by: Vi(k)

(10.71)

= /Lis

In the case of an Erlang-m model, vi(k) is given by: n

Vi(k)

=

Y-1 ‘%d”)

Pi

.

(,i,

+ Jgl(-l)‘Ei;k(J)

n

v@sl)) fii

(10’72)

’

with: Y-1 -&k(j)

=

c

z,,,(j),

T,” j h=k-j

n=j

(h, h-1,. . . , &c-j) : b’k E {k,. . . , k - j}, k o
%,h

=

1 + (bi,h,

c h=k-j (pi,h

=

Xi(h),

vh=n-j

>

,

y=min{m,k}.

Pi

For the Cox-2 model (c& 2 0.5), an even simpler equation exists: (10.73)

b(k) - L&l . bill

+ fiil

- ,%2

vi(k) = (Xi(k) + /Gil + /Liz) - vi(k - 1) ’

for k > 1.

(10.74)

458

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

We do not discuss the algorithms for the computation of the conditional throughputs of the other model type (c& < 0.5). The interested reader is referred to the original literature [Mari80]. The computed vi(b) are used in the next iteration step as load-dependent service rates for the nodes in the substitute network, Eq. (10.64). To stop the iteration, two conditions must be fulfilled. First, after each iteration we check whether the sum of the mean number of jobs equals the number of jobs in the network:

i=l

(10.75)

< E,

K where E is a suitable tolerance and K Ki = ~hri(k) k=l

is the mean number of jobs at the ith node. Second, we check whether the conditional throughputs consistent with the topology of the network:

of each node are

1 N rj - -N c ri i=l

<E

for j=l,...,N,

(10.76)

1 N N c ri i=l

where: xj r.?=-=-ej

1 ej

is the normalized throughput of node j and ej is the visit ratio. Usual values for E are 10e3 or 10b4. Thus, the method of Marie can be summarized in the following four steps: Initialize. Construct a substitute network with the same topology as the original one. The load-dependent service rates pi(k) of the substitute network for i = 1,. . . , N and k = 0,. . . , K are directly given by the original network. A multiple server node with service rate ,QL~ and rrzi servers can be replaced by a single server node with the following load-dependent service rate (see also Section 6.5): for Ic = 0, “(Ic)=

?. i Ziin{i, 7rn,}-pi,

for Ic > 0 .

NON-EXPONENTIAL

DISTRIBUTIONS

459

With Eq. (10.60), determine for i = 1,. . . , N and k = 0,. . . ,(K-1) ependent arrival rates Xi(lc) by analyzing the substitute network. Here any product-form analysis algorithm from Chapter 8 can be used. For k = K we have: Xi(K) = 0. Analyze each individual node of the non-product-form network oad-dependent arrival rate X;(k). Determine the values vi(k) and the state probabilities ri(k). The computation method to be used for the vi(k) depends on the type of node being considered (for example Eqs. (10.73) and (10.74) for Cox-2 nodes), while the ~i (k) can all be determined using Eqs. (10.62) and (10.63). Check the stopping Conditions (10.75) and (10.76). If the conditions are fulfilled, then the iteration stops and the performance measures of the network can be determined with the equations given in Section 7.2. Otherwise use Eq. (10.64), pi(k) = v,(k), t o compute the new service rates for the substitute network and return to Step 2.

Fig. IO.16

Closed queueing network.

Example 10.8 In this example we show how to apply the method of Marie to a simple closed queueing network with N = 3 nodes and K = 2 jobs (see Fig. 10.16). The service strategy at all nodes is FCFS and the service times are generally distributed with the mean service times: -

1

= 2seq

Pl

-

1

= 4sec,

P2

-

1

= 2 set,

cl3

and the squared coefficients of variation are: 2 CBl

1,

2 cf32 = 4,

Ci3 = 0.5.

el = 1,

e2 = 0.5,

e3 = 0.5.

=

The visit ratios are given as:

We assume a tolerance of E = 0.001. The analysis of the network can then be done in the given four steps.

460

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

Initialize. The service rates of the substitute network can directly be taken from the given network: /-Q(l)

= Pl(2)

=

0.5,

/42(l)

= 0.25,

= p2(2)

,~(l)

= ~~(2) = 0.5.

We obtain the substitute network from the original model by replacing the FCFS nodes with generally distributed service times by FCFS nodes with exponentially distributed service times. Compute the load-dependent arrival rates Xi (k) for i = 1,2,3, and for k = 0, 1,2: When using the convolution method (see Section S.l), we first need to determine the normalizing constants of the substitute network: G(0) = 1,

G(1) = 5,

G(2) = 17,

and the normalizing constants GE’ for the network with the short-circuited node i with Eq. (8.11) are:

GE)(O)= 1, G;h)

= -9 3

G$‘2) = 2,

GE)O) = 1, G$)

= -7 3

Gg’2) = 1,

The throughput in the short-circuited network is then given by Eq. (8.14):

G~)O) = 1, ($1)

4 = -,

12 Gj;3)2) = -*

node with (K-k)

jobs in the substitute

Now the load-dependent arrival rates can be determined using Eq. (10.60): G;)(l) Xl(O) = X(l)(2) = el = 0.429. G(l) N (2) C

Similarly, the values for the other arrival rates can be determined: X2(O) = 0.214,

X3(O) = 0.167,

X1(1) = 0.333,

X2(1) = 0.167,

h(2)

x2(2>

As(l) = 0.125, A,(2) = 0.

= 0,

= 0,

Perform the isolated analysis of the nodes of the given network. Because c& = 1, the service time of node 1 is exponentially distributed (product-form node) and we have to use Eq. (10.64) vi(lc) = pi(k). The state

NON-EXPONENTIAL

probabilities

DISTRIBUTIONS

461

for node 1 can be determined using Eqs. (10.62) and (10.63):

1

v(o) =

Xl (0)

7rr(l) = 7rr(O) * Pi(l)

= 0.412,

= 0.353,

7rr(2) = 7rl(O) * fi x1(J) j=O/Ll(j+l)

=0.235*

Nodes 2 and 3 are non-product-form nodes. Because c2si 2 0.5, for i = 2,3, we use the Cox-2 distribution to model the non-exponentially distributed service time. The corresponding parameters for node 2 are given as follows: 1 = 0.125. a2 = 2cg,

= 0.0625,

jk?l = 2p2 = 0.5, For node 3 we get: b31

= @32 = a3 =

1.

Using Eqs. (10.73) and (10.74), we can determine the conditional throughputs: v2(1)

=

X2(1&21b2

+ b21P22

X2(l)+riL22

(X2(2)

=

+

LT

+ a2lP21

h(2)jhb2 v2(2)

= ()

357

+ fiaG22

&q + /r&9)

-

v2(1)

= o.152*

Then: v3(1)

For the state probabilities

0.471,

z+(2) = 0.654.

we use Eqs. (10.62) and (10.63) and get:

7r2(0) = 0.443, 7r3(0)

=

= 0.703,

7r2(1) = 0.266, ~~(1) = 0.249,

7r2(2) = 0.291, x3(2) = 0.048.

Check the stopping condition, Eqs. (10.75) and (10.76):

i=l

k=l

K

= 7.976 - 10-3.

Because the first stopping condition is not fulfilled, it is not necessary to check the second one.

462

ALGORITHMS FOR NON-PRODUCT-FORM NETWORKS

Now we compute the new service rates of the substitute network using Eq. (10.64): pi(k) = v,(k),

for i = 1,2,3,

and go back to Step 2. Determine the load-dependent arrival rates using Eq. (10.60) and analyze the nodes of the network isolated from each other:

After four iterations we get the following final results: Marginal probabilities: ~~(0) = 0.438, 7rl(l) = 0.311, 7rl(2) = 0.251,

~~(0) = 0.438, ~~(1) = 0.271,

n3(0) = 0.719, rg(l) = 0.229,

7r2(2) = 0.291,

n,(2) = 0.052.

Conditional throughputs: ~~(1) = 0.356, Utilizations,

z+(l) = 0.466,

V2(2) = 0.151,

~~(2) = 0.652.

Eq. (7.19): pl = 1 - ~~(0) = 0.562,

p2 = 0.562,

p3

= 0.281.

Mean number of jobs, Eq. (7.26): Fl = ~~(1) + 2 .7rr(2) = 0.813,

E2 = 0.853,

x3 = 0.333.

In most cases the accuracy of Marie’s method has proven to be very high. The predecessor of Marie’s method was the iterative method of [CHW75a], which is a combination of the FES method and a recursive numerical method. A network to be analyzed is approximated by K substitute networks where the kth substitute network consists of node Ic of the original network and one composite node representing the rest of the network. The service rates of the composite nodes are determined using the FES method under the assumption that the network has a product-form solution. Then, the K substitute networks are analyzed using the recursive numerical method. The performance measures obtained from this analysis are approximate values for the whole network. If the results are not precise enough, the substitute networks are analyzed again with appropriate modifications of the service rates. These results are then the approximate values for the given network. The results thus obtained are normally less accurate than the ones derived by Marie’s method.

NON-EXPONENTIAL

463

DISTRIBUTIONS

10.1.4.3 Extended SUM and BOTT The summation method and the bottapprox method can easily be extended to non-product-form queueing networks with arbitrarily distributed service times. For the mean number of jobs K; in a -/G/l-FCFS node, we use Eq. (6.54): (10.77) Again we introduce a correction factor in Eq. (10.77) as we did in the productform case, and we get: (10.78) For multiple server nodes the mean number of jobs ??i in a -/G/m-FCFS node is given by: Ki=mpi+L.

The introduction

ai - Pmi. 1 - pi of the correction factor yields: Pi 1 _

(10.80)

- ai +Pm,.

K-m,--a

Kmm, ’ pi

Now we can use the SUM method for networks with these two additional node types by introducing Eqs. (10.78) and/or (10.80) in the system equation (9.23) and calculate the throughput X and other performance measures in the same way as in Section 9.2. We can also use the BOTT method if we use the corresponding formulae for the function to h(K;) for the bottleneck ??i = zbott. For the node type -/G/l-FCFS, /L(K~,,~,) has the form: mbott)

=

-Cl + b ’ ??;bott) + d(l+

b * Kbott,j2 + 4 - Kbott . (a - b) 7 2. (a - b)

with b = (K - 1 - a)/(K - 1). For the node type -/G/m-FCFS, the function h(Kbott) cannot be given in an exact form and therefore an approximation is used:

h(Kb,,,)= b- dc 2 +f -mbott ’ with: f= b = zbott

K-m

. f -t mbott

bott

-

a 7

K

-

a=---

1+

cg 7

mbott

+ Pmbott

+ a,

c = b2

f(4

a mbott

* zbott

’ f>s

The SUM and BOTT methods for non-product-form queueing networks can also easily be extended to the multiclass case as it was done in Sections 9.2 and 9.3 for product-form queueing networks.

ALGORITHMS FOR NON-PRODUCT-FORM

464

10.1.5

Closing

Method

for Mixed

NETWORKS

Networks

In [BGJ92], a technique is presented that allows every open queueing network to be replaced by a suitably constructed closed network. The principle of the closing method is quite simple: The external world of the open network is replaced by a -/G/l node with the following characteristics: 1. The service rate pL, of the new node is equal to the arrival rate X0 of the open network. 2. The coefficient of variation cg, of service time at the new node is equal to the coefficient of variation of the interarrival time of the open network. 3. If the routing behavior of the open network is specified by visit ratios, then the visit ratio coo of the new node is equal to 1. Otherwise the routing probabilities are assigned so that the external world is directly replaced by the new node. The idea behind this technique is shown in Figs. 10.17 and 10.18.

Fig. 10.17

Open network before using the closing method.

A very high utilization of the new node is necessary to reproduce the behavior of the open network with adequate accuracy. This utilization is achieved when there is a large number of jobs K in the closed network. The performance measures are sufficiently accurate after the number of jobs in the network has passed a certain threshold value Ki (see Fig. 10.19). Depending on the chosen solution algorithm, the following values for the number of jobs are proposed (see [BGJ92]): K = 100 K = 5000 K = 10000

mean value analysis, summation method, SCAT algorithm.

To check whether the number of jobs K in the closed network is sufficiently large, we can repeat the closing method with a larger number of jobs K and compare the results.

NON-EXPONENTIAL

Fig. IO.18 method.

Closed queueing network with the additional

DISTRIBUTIONS

-/G/l

465

node for the closing

Performance measure 1

Exact value

Fig. IO.19

Threshold in the closing method.

The closing method can be extended to multiple class queueing networks and mixed queueing networks. In the case of multiple class networks, the service rate poor of the new node is given given by R . Xcr, with r = 1, . . . , R, where R is the number of classes. In the case of mixed queueing networks, the new node is added to the open classes only and then we can use any method for closed queueing networks also for mixed queueing networks. Open nonproduct-form networks are easier to analyze using the methods introduced in Section 10.1.3. But the closing method in combination with the SCAT algorithm or the SUM method is an interesting alternative to the MVA (see Section 8.2) for mixed queueing networks in the product-form case. The main advantage of the closing method is the ability to solve mixed non-product-

form-queueing networks.

466

ALGORITHMS FOR NON-PRODUCT-FORM NETWORKS

Example 10.9 In this first example, the closing method is shown for an open product-form queueing network with Xe = 5, p12 = 0.3, p13 = 0.7, and ~21 = ~31 = 0.1 (see Fig. 10.17). The service rates are ~1 = 7, ,Q = 4, and ~3 = 3. Using Jackson’s method for this network, we obtain the following values for the mean response time T and the mean number of jobs in the open network:

In Table 10.12 the values for these performance measures are shown as a function of &lo&, the number of jobs in the closed network. The results are obtained using MVA. From the table we see that we obtain exact results if Table 10.12 Mean response time ?; and mean number of jobs ?? for several values of the number of jobs in the closed network K closed

T-

10 20 30 50 60 100

K

0.95 1.24 1.29 1.30 1.30 1.30

4.76 6.17 6.47 6.52 6.52 6.52

K&,sed is large enough and that, in this case, K&s&

= 50 is the threshold.

Sink Fig. 10.20

Open queueing network.

In this second example, the closing method is used for Example 10.10 an open non-product-form network (see Fig. 10.20) with Xe = 3 and cg = 1.5. The routing probabilities are given by: Pl2 = Oe3,

P13 = O-7,

P33 = 0.1,

p34 =

0.9,

p22

= 0.3,

p41 = 0.4.

$I24 = 0.7,

NON-EXPONENTIAL

DISTRIBUTIONS

467

Table 10.13 Input parameters for the network of Fig. 10.20

1 2 3 4

9 10 12 4

0.5 0.8 2.4 4

The other parameters are listed in Table 10.13. With the method of Pujolle (see Section 10.1.3), we can analyze the open network directly to obtain: T = 4.43 and K = 13.47. The alternative method we use to solve the network of Fig. 10.20 is to first use the closing method and then use the SUM method (see Section 10.1.4.3) to approximately solve the resulting closed non-product-form network. The results are shown in Table 10.14. From this table we see that the threshold for Table 10.14 Mean response time T and mean number of jobs K as a function of K closed K closed 100 200 500 1000 5000 10000

T

Fz

3.94 4.16 4.29 4.34 4.37 4.37

11.83 12.49 12.88 13.02 13.12 13.12

K closed is about 5000 and the results of the closing method and the method of Pujolle differ less than 3%. The open networks of Example 10.9 and Example 10.10 Example 10.11 are much easier to analyze using the Jackson’s method and the Pujolle method, respectively, rather than by the closing method. Mixed product-form networks are easy to analyze using the MVA for mixed networks but also using SCAT or SUM in combination with the closing method, especially for networks with multiple server nodes. But the main advantage of the closing method is the solution of mixed non-product-form queueing networks. As an example we use a mixed network with N = 5 nodes and R = 2 classes. Class 1 is closed and contains K1 = 9 jobs (see Fig. 10.21). Class 2 (see Fig. 10.22) is open with arrival rate Xer = 5 and the squared coefficient of variation c& = 0.7.

468

ALGORITHMS FOR NON-PRODUCT-FORM

Fig. 10.21

Closed class 1 of the mixed network.

I

I -_____-_____--______-------62--KFig. 10.22

NETWORKS

~---..-.---.---------.-------’

Open class 2 of the mixed network with an additional

There is no class switching in this network. two classes are as follows:

= 0.7,

P31,l

=

p42,l

=

P53,l

=

The routing probabilities

p24,l

= 0.3,

p35,1

=

p23,2

= 0.8,

p30,2

=

0.7,

1.0,

p42,2

=

1.0,

1.0,

p53,2

=

1.0.

0.7,

of the

Class 2: p12,2 = 1.0,

Class 1: ~~2,~ = 1.0, p23,l

node.

0.3,

p24,2

= 0.2,

p35,2

=

0.3,

The other input parameters are listed in Table 10.15. Now we solve the equivalent closed queueing network with an additional node (CL, = 5, c& = 0.7) for class 2 using the SUM method (see Section 10.1.4.3) and K2 = 500. The results for the utilization pi and the mean number of jobs Ki are listed in Table 10.16. Since SUM is an approximation method, we obtain deviations from discrete-event simulation results, which are about 6% for the utilizations pi and about 5% for the mean number of jobs Ki.

DIFFERENT SERVICE TlMES AT FCFS NODES

Table 10.15 Input parameters mixed queueing network Nodes

Pi,1

4 4 6 4 5

1 2 3 4 5

Table 10.16 Utilizations discrete-event simulation

1 2 3 4 5

10.2

= Pi,2

= cf2

the

mi

3 4 3 2 2

0.4 0.3 0.3 0.4 0.5

p and mean number of jobs from (DES) and the closing method (CLS)

0.89 0.90 0.85 0.46 0.46

DIFFERENT

ctl

for

469

0.84 0.85 0.80 0.43 0.43

4.9 5.5 4.0 1.1 1.1

5.2 5.8 4.1 1.0 1.0

SERVICE TIMES AT FCFS NODES

If the given non-product-form network is a “small-distance” away from a product-form network, we may be able to modify the MVA algorithm to approximately solve the network. Based on this idea [Bard791 extended the MVA to priority scheduling as well as to networks with FCFS nodes and with different service rates for different job classes. The key idea for treating the latter case is to replace the MVA Eq. (8.31) by: T+(k)

R 1 _ = b + c his . Kis(k - I~). ir s=l

(10.81)

This equation can also be extended for multiple server nodes [Hahn881 as: Fir(k)

= $

a (10.82) m,--2

+l

C(?rLi-l-j) i-4 j=o

with: 1 ;=

R Ai, -.cs=l Ai

1 t%s ’

470

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

In order to demonstrate the use of this method, consider a queueing network consisting of N = 6 nodes and R = 2 job classes. Nodes 1 to 5 are FCFS nodes with one server each and node 6 is an IS node. Class 1 contains Ki = 10 jobs and class 2 has K2 = 6 jobs. All service times are exponentially distributed. Table 10.17 shows all service rates and visit ratios for the network. Service Tab/e 10.17 Parameters described in Fig. 10.22 i

PLzl

l-422

1 2 3 4 5 6

* 1 1 1 1 0.1

0.5 1 1 1 1 lo7

of the network

61

8 2 2 2 2 1

ei2

20 2 4 6 8 1

rate ~11, marked with an * in the table, is varied and the mean overall response time for all class-l jobs is determined. We compare the results of the extended MVA with DES. Tab/e10.18 Mean overall response time of a class-l job for different service rates ~11 Pll

MVA-Ext DES

.

0.5

2

128

260.1 260.1

143.1 141.1

105.4 102.0

The values given in Table 10.18 show that the extension of the MVA by Eq. (10.81) gives good results for queueing networks with FCFS nodes with different service rates for different job classes. SCAT and Bard-Schweitzer algorithm can also be extended [F&+39] in a similar fashion.

10.3

PRIORITY

NETWORKS

We will consider three different methods for networks with priority classes: Extended MVA, shadow server technique and extended SUM method. 10.3.1

Extended MVA (PRIOMVA)

If a queueing network contains nodes where some job classes have priority over other job classes, then the product-form condition is not fulfilled and the techniques presented in Chapters 8 and 9 cannot be used. For the approximate analysis of such priority networks [BKLC84] suggest an extension of the MVA.

PRIORITY NETWORKS

471

The MVA for mixed queueing networks is thus extended to formulae for the following priority node types: l

-/M/l-FCFS

PRS (preemptive resume):

As per this strategy, as soon as a high priority job arrives at the node it will be serviced if no other job with higher priority is in service at the moment. If a job with lower priority is in service when the higher priority one arrives, then this low priority job is preempted. When all jobs with higher priority are serviced, then the preempted job is resumed starting at the point where it was stopped. It is assumed that the preemption does not cause any overhead. The service strategy within a priority class is FCFS. l

-/M/l-FCFS

HOL (non-preemptive head-of-line):

Here a job with higher priority must wait until the job that is processed at the moment is ready to leave the node even if its priority is lower than the priority of the arriving job. The service strategy within a class is again FCFS. The priority classes are ordered linearly where class R has lowest priority and class 1 has the highest one. At first we consider -/M/l-FCFS PRS nodes and derive a formula for the mean response time T,. In this case the mean response time of a class-r job consists of: l

The mean service time:

l

The time it has to wait until all jobs with higher or the same priority are serviced:

r KS c-a s=l

l

PL,

The time for serving all jobs with a higher priority the response time Fr :

that arrive during

r-1 T, ’ A, . c PS s=l

If we combine these three expressions, then for the mean response time of a priority class-r job, 1 5 T 5 R, we get the following implicit equation:

T,$K”+C-+L, r-1 T, *A, s=l

PS

s=l

A

Pr

472

ALGORITHMS FOR NON-PRODUCT-FORM NETWORKS

where K, denotes the mean number of class-s jobs at the node. With the relation ps = X,/p3 it follows that:

(10.83)

If, on the other hand, we consider an -/M/l-FCFS HOL node, then the considered job can not be preempted any more. In this case the mean response time consists of: l

The mean service time:

l

The time needed to serve higher priority waiting time with the mean W =

l

The mean time needed for the job that is being served at the moment:

jobs that arrive during the

cRPs s=l

l

L’

The mean service time of jobs with the same or higher priority that are still in the queue:

cr K-P,. s=l

PS

If we combine these three terms for the mean response time of a priority class-r job, 1 5 T 5 R, we get:

rz-ps

Tr=C_+qf+C s=l

Rp s=l

r-l Tr-i s=l

.$+

(

)

If we use the relation ps = Xs/ps, then we get: 07 2+ c s=l Ps

e s=r+lPs r-l

CCPS

s=l

ps .

(10.84)

PRIORITY NETWORKS

473

The equations just developed give the exact T, values for M/M/lpriority queues. But in a priority queueing network the arrival process at a node is normally not Poisson. This condition means that if we apply the equations given to a priority network, we do not get exact results. Furthermore, these equations allow class-dependent service time distributions, while in product-form queueing networks a class-independent service time distribution is required at FCFS nodes. Nevertheless the approximation of [BKLC84] is based on these assumptions. Let xi,” denote the mean number of class-s jobs at node i that an arriving class-r job sees given the network population vector k. If Eq. (10.83) is applied to node i of a priority network, then this equation can be generalized as follows:

(10.85) 1 - c Pis s=l

Similarly, Eq. (10.84) is of form:

T;,.(k)= & +

r i+;‘(k) s=l

s=r+l

PiS

Pis

r-l 1 -

For the computation of $,T’(k), to hold:

++

c-

c s=l

(10.86)

Pi.3

the arrival theorem (see [BKLC84]) is assumed

wj,“(k) = K. eas(k

- l,),

for closed classes r, for open classes r.

&s(k),

For open class s, the equation: -

KiS(k) = A,* ci,.Ti,(k)

(10.87)

(10.88)

can be used to eliminate the unknown Ei, (k). Then in Eqs. (10.85) and (10.86) we only need to determine the utilizations pis. For open classes we have pis = &slPis, independent of the number of jobs in the closed classes. But for closed class s the utilization pis depends on the population vector and it is not obvious which pi,(k), 0 5 k 5 K, should be used. However, [ChLa83] suggest: Pis

=

pis

(k

-

K;,),

(10.89)

where (k - Kis) is -the population vector of the closed classes with Ki, jobs fewer in class s. If Ki, is not an integer, then linear interpolation is used to

474

ALGORITHMS FOR NON-PRODUCT-FORM NETWORKS

get a convenient value for pis. Equation (10.89) is based on the assumption that if there are already Ei, jobs at node i, then the arrival rate of class-s jobs at node i is given by the remaining k - Ki, jobs in the network, and hence: p

= hs(k - ;i?is)

is

=

pis(k

- Xi,).

PiS

If we insert Eqs. (10.87)-(10.89) into Eqs. (10.85) and (10.86), respectively, then the mean response time of the closed class r at the priority node i when there are k jobs in the closed classes of the network:

PRS:

T+(k)

= 1 -

sgl

ds

r &(k HOL:

- 1,)

+

5

pis(k

-

1,)

c

1 = -+

Ti,(k)

(10.90)

,

r-l

s=l

s=r+l

/JJiS

Pis

r-l

f%r 1 -

c

(10.91)

P:,

s=l

where: p, =

pis(k - Ki,),

for closed class s, for open class s.

is PiS,

Analogously, for the mean response time of the open class r at the priority node i we get:

PRS: Fir(k) =

(10.92) 1- c

s=l

HOL:

T+(k)

Pi,

=

r-l

l-

c

s=l

Pls

Now the modified MVA with the additional priority node types can be given. The closed classes are denoted by 1, . . . , CL and the open classes by 1 * * >OP. Because of the fact that the values of l?is (k) are also needed for the open class s, it is necessary to determine the performance measures for the open classes in each iteration step of the algorithm. Initialize. For each node i = 1, . . . , N determine the utilizations of the open classes op = 1, . . . , OP using Eq. (8.55), that is: Pi,op =

1 -X0P%,0p7 Pi,op

PRIORITY NETWORKS

and check for stability Set Ki,,l(O) 1?* - - ?CL.

&,

475

5 1).

= 0 for all nodes i =

1,

. . . , N and all closed classes cl =

Construct a closed model that contains only jobs of the closed classes and solve this model using the MVA. Iterate for k = 0,. . . , K: I] r = l,... [] (8.46).

Determine Ti, (k) for all nodes i = 1, . . . , N and all closed classes , CL using Eqs. (8.53), (8.54), and (10.90) and (10.91), respectively. Determine X,(k) for all closed classes r = 1, . . . , CL using Eq.

-22.31 Determine ??i, (k) for all nodes i = 1, . . . , N and for all closed classes r = l,... , CL using Eq. (8.47). [] Determine ??ir (k) for all nodes i = 1, . . . , N and all open classes r = l,... ,OP using Eqs. (8.51) and (8.52), and Eqs. (10.88), (10.92) and (10.93) respectively. Determine

all other performance measures using the computed

results. The computation sequenced from the highest priority down to the lowest one. The algorithm is demonstrated in the following example: Example 10.12 Consider the queueing network in Fig. 10.23. The network consists of N = 2 nodes and R = 3 job classes. Class 1 has highest priority and is open, while the classes 2 and 3 are closed. Node 1 is of type

----

Source

w

Fig. 10.23

-

Mixed

network

with

Open Class Closed Class

priority

nodes.

-/G/l-PS and node 2 is of type M/M/l-FCFS PRS. The arrival rate for class1 jobs is Xi = 1. Each closed class contains one job (K2 = Ka = 1). The

476

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

mean service times are given as: -

1

-

= 0.4,

Pll

-

1

= 0.6,

P21

1

= 0.3,

-

1

P12

Pl3

1 = 0.5, CL22

-

= 0.5,

1

=

0.8.

p23

The visit ratios are: ell = 2,

el2

=

1,

el3

1,

=

e21

=

1,

Initialize. Determine the utilizations ~11 = hell+ L

= 0.8,

e22 = 0.5,

e23 = 0.4.

of the open class:

~21 = JW21.

=0.6.

k

Pll

Set Frz(O)

= 7713(O)

=

FT22(0)

=

= 0.

X23(O)

Analyze the model using the MVA. MVA-iteration

for k = (0,O):

(-1 Mean number of jobs in the open class 1: Node 1, Eq. (8.51): Pll %l(O,

[l + 7G2(0,0>

+ K3(0,

O)]

= 4

0) = 1 -

Pll

Node 2, Eqs. (10.88) and (10.92): K21(0,0>

MVA-iteration

= Xl-

VP21 e21. -=& 1 - P21

for k = (0,l):

1-2.11 Mean response time for the closed class 3: We use Eqs. (8.53) and (10.90) to get:

A-+Eg?!.l Tl3(0,1)

= I*

1

CL13

[STEP-]

1 -

Throughput

= 2.5,

T23(O,

1) =

-4.25.

Pll

l-

P21

of class 3, from Eq. (8.46): X3(0, 1) = 0.238.

IJ

Mean number of jobs in class 3, from Eq. (8.47): T&3(0,1)

= 0.595,

K23(O,l)

= 0.405.

-

PRIORITY NETWORKS

477

1-1 Mean number of jobs in the open class 1, using Eqs. (8.51)) (lO.SS), and (10.92), respectively: &(o, 1) = 6.38, &(O, 1) = 1.5. MVA-iteration

for k = (1,O) :

p5TE-xq

C&(1,0)

ri+zFTq

X2(1,0) = 0.308,

-1

Kiz(l,

mj

3i;,,(l,O)

MVA-iteration

= 1.5,

%(l,

0) = 0.462, = 5.85,

0) = 3.5,

?&(l,

0) = 0.538,

K&,0)

= 1.5.

%(l,l)

= 3.5,

for k = (1,l):

piTiF%q

T&l)

= 2.393,

[I

X2(1, 1) = 0.241,

[I

&(l,

1) = 0.578,

z&l,

[I

FIs(l,

1) = 3.654,

??&l,

1) = 4.923,

1) = 0.422,

11

X3(1, 1) = 0.178,

jJ

3T&l)

= 0.65,

1?&1)

= 0.35,

[I

K&l)

= 8.91,

17&l)

=1.5.

These are the final results for the given mixed priority

network.

Up to now we considered a pure HOL service strategy or a pure PRS service strategy at a node. In many cases we may need to combine these two priority strategies. Let us therefore consider a combination of PRS and HOL: HOL-group

Fig. 10.24

I

Splitting

PRS-group

into HOL and PRS classes.

In this combination, some job classes are processed by HOL and some job classes are processed by PRS. It is assumed that HOL jobs always have higher priority than PRS jobs (see Fig. 10.24). In this case:

478

ALGORITHMS FOR NON-PRODUCT-FORM 0

NETWORKS

An arriving HOL job cannot preempt a HOL job that is still in progress, but it can preempt a PRS job that is still in progress. Therefore the response time of a HOL job is not influenced by the arrival of PRS jobs. r K-P,

T,=c,-

+k;+@-;).2+;.

s=l

s=l

s=l

By simplifying the above expression we get, in analogy to Eq. (10.91): %~glp) Tir(k)

= ;

+ s=$+l P&t-yL)

+ ‘=’ 1 -

j&s

0

=

pi,&

-

,

r-l c s=l

(10.94)

As

%s).

A PRS job can always be preempted by a job with higher priority. priority class r we therefore get: r-1 T, -A,

T,,f++c-+A. s=l

For

PS

PS

s=l

Pr

In analogy to Eq. (10.90) we get after simplification:

(10.95)

@is

=

pi+

-

Eis).

If these formulae for Tir (k) are used in the MVA instead of Fir, then we get an approximate MVA algorithm for closed queueing networks with a mixed priority strategy. In the next section we introduce the technique of shadow server that can also be used to analyze priority networks. 10.3.2

The Method

of Shadow

Server

10.3.2.1 The Original Shadow Technique This technique, shown in Fig 10.25, was introduced by [Sevc77] for networks with a pure PRS priority strategy. The idea behind this technique is to split each node in the network into R parallel nodes, one for each priority class. The problem then is that in this extended node, jobs can run in parallel whereas this is not possible in the original node. To deal with this problem and to simulate the effect of preemption, we iteratively increase the service time for each job class in the

PRIORITY NETWORKS

479

shadow node. The lower the priority of a job, the higher is the service time for the job. After all iterations we have: Si,r

L

Si,r*

Here 5i,r = l/pir denotes the approximate service time of class-r jobs at node i in the shadow node. Class 1 rfut-eClass i

cL

--g--@-

Original

Class R

l

’

IIIIIGShadow Node

Node

Fig. 10.25

-J-j-@-

. .

The idea behind the shadow technique.

Shadow Algorithm

Transform the original model into the shadow model. Set X+ = 0. Iterate: 1-1

Compute the utilization e Pi,r

(1

for each shadow node:

=

Xi,r

(10.96)

’ i?i,r.

Compute the shadow service times: Si,r

Si,r

=

r-l l

-

C s=l

'

(10.97)

&,s

Here si,r denotes the original service time of a class-r job at node i in the original network while s”i,r is the approximated service time in the shadow network. 1-1 Compute the new values of the throughput of class r at node i, Xi,r, of the shadow model using any standard analysis technique from Chapters 8 and 9. If the Xi,, differ less than E in two successive iterations, then stop the iteration. Otherwise go back to Step 3.1

ALGORITHMS FOR NON-PRODUCT-FORM

480

10.3.2.2

NETWORKS

Extensions of the Shadow Technique

Measurements on real systems and comExtension of Kaufmann 10.3.2.2.1 parisons with discrete-event simulation results have shown that the technique of shadow server is not very accurate. Because of the lack of accuracy, [Kauf84] suggested an extension of the original shadow technique. This extended method differs from the original method in that an improved expression &,r is used instead of pi,f. To derive this improved utilization we define: c = class of the job under consideration, Ni,r = number of class-r jobs at node i in the shadow node, s.Pt a,r -

approximated mean service position time, r-l

1=1

For a pure PR strategy the following relations hold: l

The improved utilization

/3ir is expressed using s$,.:

/%,r = k,r /k,r

= P(K,r

‘> O>,

Xi,r

= P(Ni,l

= 0 A NQ

A Ni,r

P(Ni,r

l

* sft7

= 0 A * * * A Ni,r-1

= 0

> 0) * Pi,r

= P(Mi,r

= 0 A Ni,r

> 0) * pi,r,

> 0) = P(Mi,r

= 0 A Ni,r

> 0) * pi,r

* of’,:.

The last equation can now be used to derive an expression for s$: Pt ‘i,r

-

P(Ni,r P(Mi,r

> 0)

= 0 A Ni,r

> 0)

* IL”’

1 =

l

By using conditional probabilities into:

the last expression can be rewritten

l

Now we rewrite the numerator of the last expression and define the correction factor Si,r which we use to express /3i,r: 1 p

i,r

=

1 -

P(Mi,r

> 0 A C # T 1 Ni,r

> 0)

’ “,”

PRIORITY NETWORKS

481

With: bi,r = 1 - P(A&,r > 0 A c # T ] N+. > 0) and: j4,r

= h,r

bi,r

’ gi ,7-7

= Xi ,r * SPt 2.,7-T

we obtain:

Oi,r

(Ji,r

As we see, the improved utilization &,r is just given by dividing the utilization fii,r by the correction factor 6i,T. In [Kauf84] it is shown that the following approximation holds:

6;r = PW - &-)I + 44 *Pb-+ 1) 7 Ptr)[l - Ptr>12 + a(r> ’ Pi,r

for r = 2,3, . . . , R,

(10.99)

with: r-l

r-l

‘I

This correction factor S changes only Step 3.2 in the original shadow algorithm: Si,r Si,r

=

r-l 1 -

(10.100)

1

C s=l

6

* Pi,s i,s

If the utilizations of the shadow nodes are very close to 1, then the expression: r-l 1 -

C $ * Pi,s SE1 2,s

can be negative because of the approximate technique. In this case the iteration must be stopped or the extended shadow technique with bisection can be used. In the original and extended shadow method the expression: r-l cs=l

Pi,s

can be very close to 1. Then the service time si,r gets very large and the throughput through this node is very low. In addition, the value of: r-l cs=l

Pi,s

482

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

will be very small in the next iteration step and in some cases the shadow technique does not converge but the computed values move back and forth between the two values. Whenever such swinging occurs, we simply use the average of the last two values of throughput in Step 3.3. Thus we have used bisection. If class switching is allowed in the network then 10.3.2.2.2 Class Switching we use the shadow technique in combination with the concept of chains (see Section 7.3.6.1). The resulting algorithm can now be given in the following steps: Determine the chains in the network. Transform the original model into the shadow model. Set X+. = 0. Iterate: [I Pi,r

s”i,r

=

Xi,r

* s”i 7r *

Si,r

=

r-l 1 -

c 3~1

1 6

* Pi,3 i,s

Here si,r is the original service time of a class-r job at node i in the original network. I] Transform the class values into chain values (to differentiate between class values and chain values, the chain values are marked with *). I] Compute the performance parameters of the transformed shadow model with the MVA. If the model parameters swing, then set

and compute the performance measures with this new response time. -1 Transform chain values into class values. If the XF,, (chain value) in two successive steps differs less than E, stop the iteration. Otherwise go back to Step 4.1.

PRIORITY NETWORKS

483

HOL Group

PR Group Original

Shadow Node

Node

fig. 10.26

Shadow technique and mixed strategy.

The original shadow tech10.3.2.2.3 Class Switching and Mixed Priorities nique [Sevc77] can only be used for networks with a pure PRS-strategy, but in the models of a UNIX-based operating system (see Section 13.1.4), for example, or in models of cellular mobile networks [GBB98], a mixed priority strategy is used. For this reason the original shadow technique is extended to deal with mixed priority strategies. Consider the situation where the classes 1,2, . . . , u are pure HOL classes and u + 1,~ + 2,. . . , R are pure PRS classes, as shown in Fig. 10.26. We define: u Yi,r =

c

Ni,l,

l=l,l#r

and get the following relations: The utilizations

l

can be expressed as: n Pi,r =A. v- .sPt z,r) fii,r = P(Ni,r >

0).

l

A PRS job can only be served if no other job with higher priority waiting:

l

For HOL jobs, two cases have to be distinguished:

is

ALGORITHMS FOR NON-PRODUCT-FORM NETWORKS

484

1. Class-r jobs as well as HOL jobs are at the node and a class-r job that is just served. Then we get: N,r > 0

A

c= r

A

yi,r > o,

or 2. Class-r jobs but no HOL jobs are waiting for service at a node. Then we get: Ni,r > 0

A

yi,r = 0.

For the HOL classes we therefore get: - The utilization of node i by class-r HOL jobs is given by the probability that at least one class-r job is at node i (and is served) and other HOL jobs are waiting, or at least one class-r job is at node i and no other HOL jobs are waiting for service: h,r

= Pi,r * /b,r > 0 A C = r A yi,r > 0) + P(Ni,r > 0 A yi,r = O)] * pi,r = [P(Ni,r > 0) - P(Ni,r > 0 A yi,r > 0 A c # T)] * pi,rs

= [P(n;i,r

- As in the case of a pure PRS strategy, we get analogously: P(Ni,r

sPt zz a,r

P(Ni,r

=

1-

> 0) - P(Ni,T

> 0) >

0A

-3 yi,r

>

P((yi,r >O/IL#T)ANi,r P(Ni,r

0 /I

c

# r) * “yr

>0)

“‘,‘*

> 0)

- The fraction in the numerator is now rewritten using conditional probabilities: = 1 - P(Yi,r l

> 0 A c # r) (

Ni,r

> 0) ’

As before, we now define the correction factor I/I~,~as:

@i,r = 1- P(yi,r >0 A c # r 1Ni,r

> O),

and get: ji,r

= 1;1- ’ &,7-e i,r

(10.101)

PRIORITY NETWORKS

485

As we see, the correction factor for the HOL case differs from the PRS case only in the summation index. We then get to the following equations:

e(r>[l - e(r)1 + PW * ix4 = @(r)[l - @(?+)I2 + p(r) ’ @i,r’

‘i”r

P(r) =

’ u Pi* r L c wr,k k=l,k#r r-l c

wr,k

=

1si k

if T E HOL, ’

LPi r wr,k

\ k=l

(10.102)

if r E PR, ’

7

si,r

u c k=l,k#r

t%,k,

if r E HOL,

r-l c k=l e(r)

&,k,

if r E PR,

+ pi,r*

Up to now we assuC/ass Switching and Arbitrary Mixed Priorities 10.3.2.2.4 med that the classes 1, . . . , u are pure HOL classes and the classes u + 1, . . . , R are pure PRS classes. In the derived formulae we can see that in the HOL case, the summation in the correction factor is over all HOL classes excluding the actually considered class. In the PRS case, the summation is over all classes with higher priority. With this relation in mind it is possible to extend the formulae in such a way that every mixing of PRS and HOL classes is possible. Let Jr be the set of jobs that cannot be preempted by a class-r job, r $ Jr. For the computation of the correction factor, this mixed priority strategy has the consequence that : l

For an HOL class r the summation is not from 1 to u but over all priority classes s E &..

l

For a PRS class r the summation is not from 1 to r - 1 but over all priority classes S E
For any mixed priority strategy we get the following formulae: Si,r si,r

=

1 - C 1 * fii,s ’ SE:E?. +i13

e(r)Cl - e(r)1 + P(r) - @W “J- = e(r)[l - e(r)12 + p(r) * @i,r ’

(10.103)

486

ALGORITHMS FOR NON-PRODUCT-FORM

c Pi,

NETWORKS

,r

=

wr,k

MET

‘h,k

=

’

%k -7 Sir

Fig. 10.27 Simple network for Example 10.13. Example 10.13 Consider the network, given in Fig. 10.27. Node 1 is a -/M/l-PRIORITY node and node 2 is a -/M/l-FCFS node. The network contains two priority job classes, Class 1: PRS, and Class 2: HOL, and class switching is allowed. The routing probabilities are given by: 0.4,

Pll,ll

=

0.5,

P11,12

P12,ll

=

0.5,

P21,ll

=

1.0,

p 12,12 = 0.4, p22,11 = 1.0.

=

p11,21 = 0.1, p12,22

= 0.1,

The other parameters are: Sll = 0.1, s12 = 0.1, K = 3, E = 0.001.

s21 = 1.0,

522 = 1.0,

Compute the visit ratios ei, in the network.

For this we use the

formula: eir = 5

2

ejsPjs,ir

j=ls=l

and get eri = 1.5, e12 = 1.0, epl = 0.15, e22 = 0.1. Determine the chains in the network and compute the number of jobs in each chain (refer to Section 7.3.6). In Fig. 10.28 the DTMC state diagram for the routing matrix for the chains is shown, and it can easily be seen that we can reach each state from every other state (in a state (i,r), i denotes the node number and T denotes the class number). Therefore we have only one chain in the network. IT ransform the network into the shadow network, as it is shown in Fig. 10.29.

PRIORITY NETWORKS

Fig. 10.28

DTMC state diagram of the routing matrix.

Originally

fig. 10.29

487

Node 1

The Shadow model for the network in Example 10.13.

The corresponding parameters are: sll = 0.1, 512

ell = 1.5,

e31 = 0.15,

e12 = 0.0,

e32 = 0.1,

= 0,

= 0, ~22 = 0.1,

S21

~31

= 1.0, e21 = 0.0,

532

= 1.0,

e22 = 1.0.

Computxe the visit ratios eTqper chain and remember that we have only one chain:

c %Then we get: eT1 =

ell +

el2

ell + el2

= 1,

e;, =

e21 + e22

ell +

el2

= -,2

3

e& =

e31 + e32

1 = -*

ell + el2

6

Compute the scale factors air. For this we use the equation:

488

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

air = ei, c eis’ SE9 and get: a11

=

ell ell +

= 1.0, el2

e21 Q21

+

= e31

a22

0,

+

ell +

= 0, e12

e21

= 0.6,

=

=

e22

e31 Q31

el2

e22

--

= e21

ai2 =

+ e32

0~32

= e31

e32

+

1.0,

e22

= 0.4. e32

Compute the service times per chain that are given by the equation:

which gives:

ST1 = Sll * 0111+ s12 * cl!12= 0.1, s& = s21 * a21 + s22 * a22 = 0.1, SGl

=

S31

* cY31 +

S32

* Ci!32

= 1.0.

Now we can analyze the shadow network given in Fig. 10.29 using the following parameters: 1 eT1 = 1.0, e;, = i, e& = 6’ sil = 0.1,

s;i = 0.1,

s& = 1.0,

a11 = 1.0,

a31 = 0.6,

a12 = 0.0,

cl!21= 0.0, cl!22= 1.0

K = 3,

N = 3,

Cl

J2

=

{2),

=

Q32

= 0.4, E = 0.001,

(1).

Start the shadow iterations. 1. Iteration: & = 0.505, XT1 = 5.049, q1 = 0.117,

& = 0.337, A;, = 3.366, g1 = 0.145,

& = 0.841, A;, = 0.841, s”zl = 1.000.

pT1 = 0.541, XT1 = 4.597,

& = 0.447, A;, = 3.065

& = 0.766, A& = 0.766,

;Tl = 0.125,

~$1~= 0.142,

s”& = 1.000.

2. Iteration:

PRIORITY NETWORKS

489

3. Iteration: & = 0.569, xT1 = 4.533, s”T1= 0.122,

pG1= 0.430, A;, = 3.022, z;I = 0.147,

pi1 = 0.756, A;, = 0.756, s;r = 1.000.

& = 0.556, A;, = 4.531, ST1= 0.124,

pal = 0.447, A;, = 3.012, s”gl = 0.144,

,o& = 0.755, A;, = 0.755, s”zl = 1.000.

/& = 0.565, Xi1 = 4.528,

& = 0.437, A;, = 3.018,

& = 0.754, A;1 = 0.754.

4. Iteration:

5. Iteration:

Now we stop the iteration because the difference between the X* in the last two iteration steps is smaller than E. Retransform the chain values into class values: Xl1 = all * XT1 = 4.528, AZ2 = o22 * A;, = 3.018, As1 = agl * A;, = 0.452, X32

=

Cl!32

* A;, = 0.301.

If we rewrite the results of the shadow network in terms of the original network, we get the following final results: Xpprox) = Xl1 = 4.528, XFprox) =

X31

= 0.452,

Xprox)

= X22 = 3.018,

AgProx) =

X32

=

0.301.

Verify the results. To verify these results we generate and solve the underlying CTMC via MOSES [BoHe96] (see Chapter 12). The exact results for this network are given by: AFact) = 4.52, $yt) = 0.45,

AFact) = 3.01, Apt)

= 0.30.

As we see, the differences between the exact MOSES results $Tact) and the approximated results X!Tpprox) are very small.

490

ALGORITHMS FOR NON-PRODUCT-FORM

NETWORKS

10.3.2.2.5 Quota Nodes The idea of quota is to assign to each user in the system the percentage of CPU power he or she can use. If, for example, we have three job classes and the quota are given by 41 = 0.3, q2 = 0.2 and q3 = 0.5, then class 1 customers get 30% of the overall CPU time, class 2 customers get 20% CPU time, and class 3 customers get 50% of the overall CPU time. In an attempt to enforce this quota, we give priority to jobs so that the priority of job class r, pri,, is: pri, = E.

(10.104)

1 CPU 1

I

fig.

10.30

Transformation of a quota node into the corresponding shadow node.

The higher the pri,, the higher is the priority of class r. As we can see, the priority of class-r jobs depends on the CPU utilization at node i and the quota qr of class T. A high priority job class is influenced only a little by lower priority job classes, while lower priority job classes are influenced much more by higher priority job classes. A measure for this influence is the weight i;l,,j, which is defined as:

(10.105)

The weights 13,,j are a measure for how much the rth job class is influenced by the jth job class. The closer these weights are to one, the higher is the influence. Since an active job influences itself directly (r = j), we have LZ~,~= 1. The mean virtual service time of a class-r job can approximately be given as: virt s . (27)

_ -

S&T-

1 - 5 pi,?-* iz;,,k lC=l

(10.106)

PRIORITY

Class

1

NETWORKS

491

1 . . . m

++I Class

4%

2

’

1

. . .

. . .

m

43

m

Class

R

1 . . . m

+--El -/M/m

Node

Shadow

Fig. 10.31

Shadow transformation

-/M/m

of a -/M/m

Node

node.

If we compare this equation with the shadow equation to compute the shadow service time, then the basic structure of the equations are the same. The idea, therefore, is to use the weights ;j,,j to include quota into the shadow technique. In order to transform a quota node with R job classesinto a shadow node, we apply the shadow technique as shown in Fig. 10.30. The resulting equation for computing the service time at each shadow node can now be given as: Si,r

if r E HOL,

1- 5 2 Si,r

=

/c=l kfr

r-l

Gi,k =

’ jji,k '

si,r _

priz prii + prif ’

(10.107) if r f PRS,

prik = E# 2,

By using the weights Gi,k, we make sure that each class gets the amount of CPU time as specified by the CPU quota. In the case of an arbitrary mixed priority strategy, we use the summation index < as defined before.

492

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

10.3.2.2.6 Extended Nodes We now introduce a new node type, the so-called extended -/M/ m node. It is very easy to analyze a regular -/M/m node since we only have to transform it into the corresponding shadow form and use the equations for -/M/ m nodes when analyzing the node. The transformation of a -/M/m node into its corresponding shadow form is shown in Fig. 10.31. The node shown in Fig. 10.32 is called extended -/M/m node. This node contains R priority classesthat are priority ordered (job class 1 has highest priority, job class R has lowest priority). The CPU can process all job classes,while at the APU (associate processing unit), only the job classes For a better overview, a queue U 7* ** 7R (1 5 u 5 R) can be processed.

Fig.

Extended

10.32

-/M/m

node (original

form).

is introduced for each job class. The result of this transformation is shown in Fig. 10.33. For example, let u = R = 3, then the job classes1, 2, and 3 can

I I

R

Class

u

Class

I Fig.

Class

10.33

Class

:

u-l

1

Extended

I

t-

:

I

-/M/ m node with priority

queues.

enter the CPU, while the APU can only be entered by class-3 jobs. Priority class-3jobs can always be preempted at the CPU, but in this case they cannot be preempted at the APU because this job class is the only one entering the APU. In general, the actual priority of a job also depends on the node it

PRIORITY

NETWORKS

493

enters. If it enters the CPU, the priority order remains unchanged (priority class 1 has highest priority and priority class R has lowest priority), while at the APU priority, class-u jobs have highest priority. Since preemption is also possible at the APUs, if we have more than one priority class, the shadow transformation can be applied to each node, from which a switch to the APU is possible. The result of this transformation is shown in Fig. 10.34.

Class

Class

R

u-l

---0 Class

1

CPU

.. l

CPU

u-l

1

--o--

Fig. 10.34

Extended

-/M/ m node (complete

shadow form).

The components APU,, . . . , APUR form the original APU in the network. Now it is possible to apply the shadow algorithm to the network transformed in this way. Because the priority order at the APU can change, we have to separate the APU iterations from the CPU iterations meaning that in one shadow step we first iterate over all CPUs and then over all APUs. Because over CPU and APU, we usually get different final of the separated iterations service times for CPUi and APUi, resulting in a so-called asymmetric node [BoRo94]. At the CPU, class-l jobs have highest priority, while at the APUs, class-u jobs have highest priority. In the shadow iteration, only Step 3.2 is to be

ALGORITHMS

494

FOR NON-PRODUCT-FORM

NETWORKS

changed as follows: Qc : &J.

Si,r

= l-

c S>C,SEE?-

c=l

R.

7”‘7

&*Pi,s’ (

An application for this node type can be found in Section 13.1.4 where it plays an important role. 10.3.3

Extended

SUM

The SUM method (see Section 9.2) can be applied to priority networks if we extend the formula for the mean number of jobs in the system, Fir of an individual priority node as we did for an FCFS node with multiple job classes. The extended method is referred to as PRIOSUM. Here we use the formulae for single station queueing systems with priorities (see Section 6.15). In this context it is convenient to consider class1 as the classwith the highest priority and class R as the one with the lowest priority. Of course, this convention leads to minor changes in the formulae. l

-/M/l-FCFS-HOL node After we have introduced multiple classesand considered the reverse order of priorities, we derive the following formulae from Eqs. (6.106) and (6.107): WiO

(10.108)

wir = (1 - air)(l - Crir+i) ’ with: r-l oar

pij

c j=l

=

7

r > 1,

(10.109)

otherwise,

0,

and the remaining service time: (10.110) We obtain the final result for the mean number of jobs xi, with: Kir

=

Pir

in node i (10.111)

+ Qir,

and the introduction of correction factors as in the FCFS case:

Fir = pir +

xi, F i+ j=l Pi, 1 - moir K-2

l-K-2

)(

mgir+l

* >

(10.112)

PRIORITY

l

NETWORKS

495

-/M/m-FCFS-HOL node In this case we also can use Eqs. (10.108) and (10.109). Together with:

(Pi>R Pij wio= -pm% mipi cj=l Lij

(10.113)

Fir = m&r + Qir7

(10.114)

and:

we obtain:

Fir

= mipir +

(

1_

K-m,-1 K-&

1_

Oir

>(

K-7&-1 ~~~~

pmi (Pi) . mifh Oir+l

>

(10.115)

a -/G/l-FCFS-HOL node Again with Eqs. (10.108) and (10.109) and the remaining

service time:

R Wio

=

i

C

(10.116)

XijO!2(Bij),

j=l we obtain: $xir Kir

=

pir

$J j=l

XijQJa(Bij)

+ 1

(

K-l-a

K-1

air

1-K-l-a >(

7 - K-1

Q’ir+l

(10.117)

>

with the second moment of the service time distribution: + 1 &2CBij) = (l-lij)2 , $j

(10.118)

c& + 1

(10.119)

a=:!. l

-/G/m-FCFS-HOL

node (10.120)

$hr Kir

=

W&r

+

1 _

K-mi-a K-m,

5 Pijaz(Bij)pij

j=l

Oir

pm% (Pi> .____ 1 _ )(

K-mi-a K-m,

Oir+l

miPi

(10.121)

ALGORITHMS

496

FOR NON-PRODUCT-FORM

NETWORKS

If preemption is allowed, we get: (10.122) Let WJ,‘:’ be the mean waiting time of a job of class r in node i if we consider only the classes with the r highest priorities. Then we get:

W(” 20

w!‘) = ” l

-/M/l-FCFS-PRS

(1 -

(10.123)

.

air+l)

node

r p.. Iv;;’ =x--$.

(10.124)

i=l

Using Eq. (10.111) and Little’s theorem we obtain: Fir = /3ir + AirWire

(10.125)

For wir we use Eqs. (10.122), (10.123), and (10.124) and introduce the correction factor:

.1

(10.126)

l

-/M/m-FCFS-PRS

node WCr)

= i0

-. p77;2i mi~ir+l

’ c j=l

Pij -7 Pij

(10.127)

with probability of waiting P?I;Li,which we get from the probability of waiting Pmi by replacing pi by gir+i, With Eq. (10.114) and Little’s theorem: Kir = mipir $

AirFir

and using Eqs. (10.122), (10.123), and (10.127)

r Fir = mif&r •t

(10.128)

7

we obtain:

SIMULTANEOUS

l

RESOURCE

POSSESSION

497

-/G/l-FCFS-PRS node With the remaining service time: (10.130) and Eqs. (10.118), factors:

l

(10.119),

(10.122), and (10.123) and the correction

-/G/m-FCFS-PRS node Using the corresponding equations

as before we get for this node type: (10.132)

.

I

(10.133) 10.4

SIMULTANEOUS

RESOURCE

POSSESSION

In a product-form queueing network, a job can only occupy one service station at a time. But if we consider a computer system, then a job at times uses two or more resources at once. For instance, a computer program first reserves memory before the CPU or disks can be used to run the program. Thus memory (a passive resource) and CPU (or another active resource such as a disk) have to be reserved at the same time. To take the simultaneous resource possession into account in performance modeling, extended queue&g networks were introduced [SMK82]. I n addition to the normal (active) nodes these networks also contain passive nodes that consist of a set of tokens and a number of allocate queues that request these tokens. These passive nodes can also contain additional nodes for special actions such as the release or generation of tokens. The tokens of a passive node correspond to the service units of a passive node: A job that arrives at an allocate queue requests a number of tokens from the passive node. If it gets them, it can visit all other nodes of the network, otherwise the job waits at the passive node. When the job arrives

498

ALGORITHMS

FOR

NON-PRODUCT-FORM

NETWORKS

at the release node that corresponds to the passive node, the job releases all its tokens. These tokens are then available for other jobs. We consider two important cases where we have simultaneous resource possession. These two cases are queueing systems with memory constraints and I/O subsystems. 10.4.1

Memory

Constraints

.. . @ T erminals

Fig. 10.35

Central-server

model with memory constraint.

In Fig. 10.35 we see the typical case of a queueing network with memory constraints where a job first needs to reserve some memory (passive resource) before it can use an active resource such as the CPU or the disks. The memory queue is shown as an allocate node. The job occupies two resources simultaneously: the memory and the CPU or memory and disks. The maximum number of jobs that can compete for CPU and disks equals the number of tokens in the passive node, representing available memory units. The service strategy is assumed to be FCFS. Such models can be solved by generating and solving the underlying CTMC (via SPN P for example). To avoid the generation and solution of resulting large CTMC, we use an approximation technique. The most common approximation technique is the flow-equivalent server method (FES) ( see Section 8.4). We assume that the network without considering the simultaneous resource possession would have product-form solution. The subsystem that contains the simultaneous resource possession is replaced by an FES node and analyzed using product-form methods. For simple systems with only one job class, the load-dependent service rates pi(k) of the FES node can be determined by an isolated analysis of the active nodes of the subsystem (CPU, disks), where the rest of the network is replaced by a short circuit path, and determination of the throughputs X(lc) for each possible job population along the short circuit. If K denotes the number of memory units,

SIMULTANEOUS

RESOURCE

then the service rates of the FES node are determined P(k) =

Je)l

POSSESSION

as follows

499

[SaCh81]:

for k = 1,. . . , K, for k > K.

X(K),

If there are no more than K jobs in the network, then there is no memory constraint, and the job never waits at the allocate node. Example 10.14 To analyze the central-server network with memory constraint given in Fig. 10.35, we first replace the terminals with a short circuit and compute the throughputs X(k) of th e resulting central-server network using any product-form method as a function of the number of jobs k in the network. Then we replace the subnetwork by an FES node (see Fig. 10.36). This reduced network can be analyzed by the MVA for queueing networks

Fig. 10.36

Reduced

network.

with load-dependent service rates or as we do it here using a birth-death type CTMC (seeSection 3.1). With the mean think time l/p at the terminals and the number of terminals M, we obtain for the birth rates: h=(M-k)p,

k=O,l,...,

K,

(10.134)

and for the death rates:

uk =

v4

i= l,...,K,

X(K),

i > K.

(10.135)

Marginal probabilities of the FES node are (seeEq. (3.11)): r(k) = n-(O)(“;ii& -

(10.136)

. I-I”, fp+ i=l

and with Eq. (3.12): .

(10.137)

500

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

Now the throughput X of the FES node that is also the throughput terminal requests is obtained by:

x = 5 n(i)b$= 5 +& + VK 5 i=l

i=l

n(i).

of

(10.138)

i=K+l

As a numerical example, assume that the mean think time l/p = 15sec, the maximum number of programs in the subnetwork is K = 4 and the other parameters are as shown in Table 10.19. We calculate the mean response time T for terminal requests as a function of the number of terminals M and, in comparison, the mean response time + for the case that the main memory is large enough so that no waiting for memory space is necessary, i.e., K 2 M. In the case with ample memory, the overall network is of product-form type and, hence, T can be determined using algorithms from Chapter 8. From Table 10.19 network Node CL3 Pl,

Parameters

CPU

Disk

89.3 0.05

for the Central-server 1

Disk

44.6 0.5

Table 10.20 we see that Q! is only terminals M is not too large.

a

2

26.8 0.3

Disk

3

13.4 0.15

good approximation if the number of

Table 10.20 Response time for the centralserver system with memory constraint M

10

20

30

40

50

?;

1.023 1.023

1.23 1.21

1.64 1.46

2.62 1.82

7.03 3.11

LF

For other related examples see[Triv82], and for a SHARP E implementation of this technique see [STP96]. A n extension of this technique to multiple class networks can be found in [Saue81], [B ran82], and [LaZa82]. This extension is simple in the case that jobs of different job classesreserve different amounts of memory. But if on the other side, jobs of different job classesuse the same amount of memory, then the analysis is very costly and complex. Extensions to deal with complex memory strategies like paging or swapping also exist [BBC77, SaCh81, LZGS84]. For the solution of multiple class networks with memory constraints see also the ASPA algorithm (average subsystem population algorithm) of [JaLa83]. This algorithm can also be extended to other casesof simultaneous resource possession.

SIMULTANEOUS

Problem SHARPE.

10.1

Verify the results

RESOURCE

POSSESSION

501

shown in Table 10.20 using PEPSY and

Solve the network of Fig. 10.35 by formulating it as a Problem 10.2 GSPN and solve it using SHARPE or SPNP. Compare the results obtained by means of FES approximation. 10.4.2

I/O Subsystems

Fig. 10.37

I/O subsystem.

The other very important case of simultaneous resource possessionoccurs in modeling an I/O subsystem consisting of several disks that are connected by a common channel and a controller with the CPU. In the model of Fig. 10.37 channel and controller can be considered as one single service station because they are either simultaneously busy or simultaneously idle. An I/O operation from/to a certain disk can be subdivided into several phases: The first phase is the seek phase when the read/write head is moved to the right cylinder. The second phase is the latency phase, which is the disk rotation time needed for the read/write head to move to the right sector, Finally we have the data transfer phase. During the last two phasesthe job simultaneously needs the disk and the channel. In the special casesof rotational position sensing systems (RPS) the channel is needed only in the last phase. Delays due to simultaneous resource possessionoccur if a disk is blocked and no data can be transmitted because another disk is using the channel. For the analysis of I/O subsystem models, different suggestions have been published. In the formulation of [Wilh77], an increased service time for the disks is computed approximately. This increased service time takes into account the additional delay for the competition of the channel. Each disk is then modeled as an simple M/G/l queueing system. To determine the disk service time, another model for the probability that the channel is free when needed is used. Hereby it is assumedthat only one accesspath for each disk exists, and a disk cannot be used by several processors simultaneously. This limitation is removed by [Bard801 and [Bard821 where the free path probability is determined using the maximum entropy principle. We consider a method for analyzing disk I/O systems without RPS, introduced in [LZGS84]. The influence of the channel is considered by an additional

502

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

term in the mean service time si of the disk. disk; is given then as: Si

=

S,i

+

Sli

+

Sti

The mean service time of the

+

(10.139)

sCi,

with: sli = mean latency time,

s,i = mean seek time, Sti

= mean transfer

time,

sci = mean contention

time.

The contention time sci is calculated iteratively. To obtain an approximate formula for s,i , we first use the formula for the mean number of customers in a -/M/l FCFS node in an open queueing network (see Section 6.3): (10.140)

K,=-J&. i

In the case of the channel, any requests ahead of a diski request at the channel must be associated with some disk other than i, so the equation is slightly modified to: zch

=

Pch

-

&h(i)

1 -

Pch

(10.141) ’

where:

Pch

=

5 i=l

(10.142)

&h(i)

is the utilization of the channel and &h(i) is the utilization of the channel due to requests from diski. Here we have assumed that the number of disks is equal to U. Because the mean channel service time is the sum of the mean latency time S,i and the mean transfer time sti , we obtain for the contention time: sci

=

Pch

1 -

&h(i) p Ch

* hi

+

(10.143)

St&

and, hence, for the mean disk service time: Si

=

S.yi +

(Sli

+

Sti)(l 1 -

-

p&(i))

Pch

(10.144) *

The measures &h(i) and &h, and other interesting performance measures can be calculated using the following iterative algorithm. We assume that the overall model is a closed queueing network that will be of product-form type if we have no contention at the channel. Initialization.

System throughput

X = 0.

SIMULTANEOUS

RESOURCE

POSSESSION

503

Iteration:

I

STEP 2 1 channel:

For each diski, the contribution

of diski to the utilization

(10.145)

pch(i) = A. ei(qi + sti). . STEP 2 2

of the

Channel utilization:

ikh

=

(10.146)

x&h(i). i=l

-9

Compute the mean service time for each diski using Eq. (10.144).

I] Use MVA or any other product-form method method to calculate system throughput X. Return to Step 2.1 until successive iterates of X are sufficiently close. Obtain the performance

measures after the final iteration.

Table 10.21 Iterative calculation of the throughput and the channel utilization. k’ch (i>

2 3 4 5 6 7 8 9 10 11 12

0 0.056 0.030 0.050 0.038 0.047 0.041 0.045 0.042 0.044 0.043 0.043

0 0.167 0.090 0.150 0.113 0.140 0.123 0.135 0.127 0.132 0.129 0.130

l’ch

0 0.836 0.449 0.749 0.564 0.701 0.611 0.674 0.635 0.659 0.645 0.651

si

s ei

11.00 23.24 12.96 18.16 14.10 16.63 14.77 15.96 15.18 15.63 15.36 15.48

x out

0.056 0.030 0.050 0.038 0.047 0.041 0.045 0.042 0.044 0.043 0.043 0.043

Example 10.15 We consider a batch computer system with multiprogramming level K = 10, the mean CPU service time of 15sec, one channel, and five disks [LZGS84], each with the mean seek time s,i = 8sec, the mean latency time sli = 1 set, and the mean transfer time sti = 2 sec. We assume ei = 1 for each node. With a queueing network having six nodes, we can solve the problem iteratively using the preceding method. The first 12 steps of the iteration are given in Table 10.21. This iterative algorithm can be easily extended to disk I/O systems with: l

Multiple classes

. RPS

504

ALGORITHMS

l

Additional

FOR NON-PRODUCT-FORM

NETWORKS

channels

0 Controllers For details see [LZGS84]. [STP96]. Problem algorithm

F or related examples

see

10.3 Verify the results shown in Table 10.21 using the iterative of this section and implement it in SHARPE.

Problem 10.4 Develop a stochastic and solve using SPN P. 10.4.3

of the use of SHARPE,

Method

of Surrogate

reward

net model for Example 10.15

Delays

Apart from special algorithms for the analysis of I/O subsystem models and queueing networks with memory constraints, more general methods for the analysis of simultaneous resource possessionin queueing networks are also known. Two general methods with which a satisfactory accuracy can be achieved in most casesare given in [FrBe83] and [JaLa82]. Both methods are based on the idea that simultaneously occupied service stations can be subdivided into two classes:the primary stations that are occupied before an access to a secondary station can be made. In the caseof I/O subsystem models, the disks constitute the primary stations and the channel the secondary station. In the case of models with memory constraints, the memory constitutes the primary stations and the CPU and disks constitute the secondary stations. The time while a job occupies a primary station without trying to get service from the secondary station is called non-overlapping service time. This non-overlapping service time can also be zero. The time while a job occupies both the primary and the secondary station is called overlapping service time. Characteristic for the method of surrogates [JaLa82] is that the additional delay time can be subdivided into: l

The delay that is caused when the secondary station is heavily loaded. Heavy load means that more jobs arrive at a station than can immediately be served.

l

The delays due to an overload at the primary station. This is exactly the time one has to wait for a primary station while the secondary station is not heavily loaded.

To estimate the delay times, two models are generated, where each model provides the parameters for the other model. In this way we have an iterative procedure for the approximate determination of the performance measuresof the considered queueing model. This method can be applied to single class closed queueing networks that would otherwise have product-form solution

if we neglect the simultaneous resource possession.Studies have shown that the method of surrogate delays tends to overestimate the delay and therefore

SIMULTANEOUS

RESOURCE

POSSESSION

505

underestimate the throughput. In all examples we considered , the deviation compared to the exact values was found to be less than 5%. 10.4.4

Serialization

Up to now we only modeled contention for the hardware resources of a computer system. But delays can also be caused by the competition for software resources. Delay that is caused because a process has to wait for software resourcesand that makes a serialization in the system necessary is called serialization delay. Examples for such software resources are critical areas that have to be controlled by semaphoresor non-reentrant subprograms, Serialization can be considered as a special case of simultaneous resource possession becausejobs simultaneously occupy a special (passive) serialization node and an active resource such as CPU or disk. The phrase serialization phaseis usedhere for the processingphase in which at most one job can be active at any time. A job that wishes to enter the serialization phase or that is in the serialization phase is called a serialized job. The processingphasein which a job is not serialized is called the non-serialized phase, and a job that is in a non-serialization phase is called a non-serialized job. There can always be more than one job in the non-serialization phase. All nodes that can be visited by serialized jobs are called serialized nodes, all other nodes are called non-serialized nodes. Each serialized node can be visited by serialized jobs as well as non-serialized jobs. An iterative algorithm for the analysis of queueing networks with serialization delay is suggested by [JaLa83]. This algorithm can also be used for multiple class networks where the jobs can be at most in one serialization phase at one time. The starting point of this method is the idea that the entrance of a job in a serialization phase shall lead to a class switch of that job. Therefore a class-r job that intends to enter the serialization phase s is denoted as a class-(s,r) job. When the job leaves the serialization phase, it goesback to its former class. In each serialization phase, there can be at most one job at a time. Non-serialization phasesare not limited in this way. The method uses a two layer model and the iteration takes place between these two layers. For more information see [JaLa83]. There are also other formulations for the analysis of queueing networks with serialization delays, but these methods are all restricted to single class queueing networks. In the aggregate server method [AgBu83], an additional node for each serialization phase is introduced and the mean service time of the node is set equal to the mean time that a job spends in the serialization phase. The mean service time of the original node and the additional node have to be increased in a proper way to take into consideration the fact that at the original node the service of non-serialized jobs can be hindered by the service of serialized jobs and vice versa. Therefore, iteration is needed to solve

the model,

506

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

Two different techniques to model the deleterious effect of the serialization delays to the performance of a computer system are proposed by [Thom83]: an iterative technique and a decomposition technique. The decomposition technique also uses two different model layers. At the lower layer the mean system throughputs for each possible system state is computed. These throughputs are then used in the higher level model to determine the transition rates between the states from which the steady-state probabilities can be determined by solving the global balance equations of the underlying CTMC. More details and solution techniques can be found in [AgTr82], [MiiRo87], and [SmBr80]. A SHARPEimplementation can be found in [STP96]. Another complex extension of the models with serialization delays are models for the examination of synchronization mechanism in database systems (concurrency control). These mechanisms make sure that the data in a database are consistent when requests and changes to these data can be done concurrently. These models can be found in [BeGo81], [CGM83], [MeNa82], [Tay87], or [ThRy85].

10.5

PROGRAMS

WITH

INTERNAL

CONCURRENCY

A two-level hierarchical model for the performance prediction of programs with internal concurrency was developed by Heidelberger and Trivedi (see [HeTr83]). Consider a system consisting of A4 servers (processors and I/O devices) and a pseudo server labeled 0. A job consists of a primary task and a set of secondary tasks. Each primary task spawns a fixed number of secondary tasks whenever it arrives at node 0. The primary task is then suspended until all the secondary tasks it has spawned have finished execution. Upon spawning, each of the secondary tasks is serviced by network nodes and finally on completion return to node 0. Each secondary task may have to wait for its siblings to complete execution and return to node 0. When all its siblings complete execution, the primary task is reactivated. It should be clear that a product-form queueing model cannot be directly used here as the waiting of the siblings at node 0 violates the assumptions. Of course, under the assumption of service times being exponentially distributed, a CTMC can be used. In such a CTMC, each state will have to include the number of tasks of each type (primary, secondaryl, secondary2, etc.) at each node. The resulting large state space makes it infeasible for us to use the one-level CTMC model even if it can be automatically generated starting from an SPN model. The two-level decomposition we use here is based on the following idea: Assume that the service requirements of the primary and secondary tasks are substantial so that on the average, queue lengths in the network will have time to reach steady state before any spawned task actually completes. We could then use a product-form queueing network model to compute throughputs and other measures for a fixed number of tasks of each type. This is assumed to be a closed product-form network and, hence, efficient algorithms such as MVA

PARALLEL

PROCESSING

507

can be used to compute steady-state measures of this lower level queueing network. A CTMC is constructed as the outer model where the states are described just by a vector containing the numbers or each type of tasks currently in the system. To describe the generator matrix of the CTMC, we introduce the following notation. For simplicity, we assume that the parent task spawns exactly two children after arriving at node 0. The parent tasks are labeled 0, while the children tasks are labeled 1 and 2. Let ai denote the number of tasks of type i. Then a generic CTMC state is a = (a~, ar,az). The state space of the CTMC will then be:

S ={a:O~a~~N,O~a~~N-ao,O~a~~N-ao,N
(a0- l,w + l,aa + 1)

x0(4

(ao,m (u0,u1,u2

X1(8)(1

(a0 (a0

+ +

-

La2)

- 1) l,Ul 1, al,

-

142) a2 - 1)

Transition

da, b)

Xa(a)(l

-n(a)) - r2(a))

Xl

(a>>

(a)@1

A:!(4 6-2(4

>

Task Task Task Task Task

0 1 2 1 2

Explanation completion, completion, completion, completion, completion,

tasks 1 and 2 spawned sibling active sibling active sibling waiting sibling waiting

implementation of this algorithm see [STP96].

10.6

PARALLEL

PROCESSING

Two different types of models of networks with parallel processing of jobs will be considered in this section. The first subsection deals with a network in which jobs can spawn tasks that do not need any synchronization with the spawning task. In the second subsection we consider fork-join networks in which synchronization is required. 10.6.1

Asynchronous

Tasks

In this section we consider a system that consists of m active resources (e.g., processorsand channels). The workload model is given by a set of statistically

508

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

independent tasks. Each job again consists of one primary and zero or more statistically identical secondary tasks. The primary task is labeled 1 and the secondary tasks are labeled 2. A secondary task is created by a primary task during the execution time and proceeds concurrently with it, competing for system resources. It is assumedthat a secondary task always runs independently from the primary task. This condition in particular means that we do not account for any synchronization between the tasks. Our treatment here is based on [HeTr82]. The creation of a secondary task takes place whenever the primary task enters a specially designated node, labeled 0; the service time at this node is assumedto be 0. Let eii, i = 1,2, . . . , m denote the average number of visits to node i per visit to node 0 by a primary task. Furthermore ei2, i = 1,2,. . . , m denotes the average number of visits to node i by a secondary task. After completing execution, a secondary task leaves the network and eir, ei2 < 00. Let l/pij denote the average service time of a type j task at node i. Each task type does not hold more than one resource at any time. The number of primary jobs in the system is assumedto be constant K and concurrency within a job is allowed only through multitasking, while several independent jobs are allowed to execute concurrently and share system resources. In the casethat the scheduling strategy at a node is FCFS, we require each task to have an exponentially distributed service time with common mean. In the case of PS or LCFS-PRS scheduling, or when the node is an IS node, an arbitrary differentiable service time distribution is allowed and each task can have a distinct service time distribution. Since concurrency within a job is allowed, the conditions for a product-form network are not fulfilled (see [BCMP75]). Th erefore we use an iterative technique where in each step a product-form queueing network is solved (the product-form solution of each network in the sequenceis guaranteed due to the assumptions on the service time distributions and the queueing disciplines). The queueing network model of the system consists of two job classes,an open classand a closed class. The open classesare used to model the behavior of the secondary tasks and, therefore, the arrival rate of the open classis equal to the throughput of the primary task at node 0. The closed class models the behavior of the primary tasks. Because there are K primary tasks in the network, the closed class population is K. Notice that the approximation assumesthe arrival process at node 0 is a Poissonprocess that is independent of the network state. Due to these two assumptions (Poisson process and independence of the network state), our solution method is approximate. A closed form solution is not possible because the throughput of the closed chain itself is a non-linear function of the arrival rate of the open chain. For the solution of this non-linear function, a simple algorithm can be used. In [HeT’r82] regu1a f aIsi was used and on the average only a small number of iteration steps was necessary to obtain an accurate solution of the non-linear function.

PARALLEL

PROCESSING

509

Let Xuz denote the arrival rate and X2 the throughput of the open class, respectively. If any of the queues is saturated, then X2 < X02. Furthermore let Xi denote the throughput of the closed class at node 0. We require X2 = Xi and for the stability of the network, we must have X2 = X02. Therefore the following non-linear fixed-point equation needs to be solved: Wo2)

=

x02.

(10.147)

Here X1(X02) is the throughput of the primary task at Node 0, given that the throughput of the secondary task is X02. Equation (10.147) is a non-linear function in X02 and can be evaluated by solving the product-form queueing network with one open and one closed job class for any fixed value of X02. Let XT and Xz, denote the solution of Eq. (10.147), i.e., Xi = X& = Xr (Xi,). Here Xi, is the approximated arrival rate of a secondary task. At this point, the arrival rate of a secondary task is equal to the throughput of a primary task at node 0. To get the approximate arrival rate X02, we can use any algorithm for the solution of a non-linear function in a single variable. Since the two classes share system resources, an increase of X02 does not imply an increase in Xi; we conclude that Xi is a monotone non-increasing function of X02. If of the open class, then the stability x yax denotes the maximum throughput condition is given by: x02 < ,yax. The condition for the existence of a stable solution, no queue is saturated, for Eq. (10.147) is given by: lim X1(X02) < AFax. x02+Xy=

(10.148) i.e., a solution where (10.149)

If a stable solution exists, it is also a unique one (monotonicity property). If the condition is not fulfilled, primary tasks can generate secondary tasks at a rate that exceeds the system capacity. The node that presents a bottleneck determines the maximum possible throughput of the open class. The index of this node is given by: (10.150) Here argmax is the function that determines the index of the largest element in a set. Therefore the maximum throughput is given by: max x2

=

Pbott,2

(10.151)

ebott,2

In Fig. 10.38 the three types of possible behaviors, depending on the system parameters, are shown. In case (a), node bott is utilized by the primary task, i.e.: ebott,l pbott,l

>

0.

(10.152)

510

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

6 i

Open

Chain

Open

Throughput

Open

Chain

Chain

Throughput

Throughput

Fig. 10.38 Three types of possible behavior of the approximation method tention at bottleneck (b) N o contention at bottleneck, moderate contention devices (c) No contention at bottleneck, little contention at other devices.

(a) Conat other

When the queue of node bott grows without bound due to an excessive arrival of jobs of the open job class, the throughput of the closed job class will approach zero. If Condition (10.152) is not satisfied, but some other network nodes are shared by two job classes,then either case (b) or case (c) results depending on the degree of sharing. Because case (b) yields a unique solution to (10.147), we can conclude that (10.152) is a sufficient but not necessary condition for convergence, while (10.150) is both necessary and sufficient. Compared to Condition (10.150), Condition (10.152) has the advantage that it is testable prior to the solution of the network. We can also extend this technique to include caseswhere more than one type of secondary task is created. Assume, whenever a primary task passes node 0, a secondary task of type k, (Ic = 2,. . . , C) is created with probability pk. To model the secondary tasks, C - 1 open job classes are used with arrival rate Xek each. Let Xe = CF=, Xek denote the total arrival rate of the secondary tasks. The individual arrival rates are constrained so that AOk = pkAO$ 2 2. The total throughput of secondary tasks is set equal to the throughput of the primary tasks at node 0. This condition again defines a non-linear equation in the single variable X. This equation is similar

PARALLEL

PROCESSING

511

(b)

Fig. 10.39 I/O (b).

The central-server

model without

overlapped

I/O (a) and with overlapped

to Eq. (10.147) and must be solved. As a concrete example we consider the central-server model. The standard model without overlap is shown in Fig. 10.39a. Node 1 represents the CPU and nodes 2, 3, 4, and 5 represent I/O devices such as disks. In the case without overlap, the number of primary tasks is constant C. The processing time of the primary tasks at the CPU is assumed to be random with mean l/pp. As soon as a primary task finishes processing, a secondary task is created with probability f. Once a secondary task is generated, the primary task returns to the CPU for processing (with mean l/pi), while the secondary task starts at the CPU queue (with mean service time l/piz). With probability ~12, the secondary task moves to an I/O device. The secondary tasks leaves the system with probability 1 - p2 and returns to the CPU with probability ~2. In the case that a primary task completes CPU processing and does not create a new secondary task (with probability 1 - f), it begins an overhead period of CPU processing with mean ~/PO. On completion of this overhead processing, the primary task moves to I/O device i with probability pi and then returns to the CPU. As scheduling strategies, we assumePS at the CPU and FCFS at all I/O devices with mean service times l/p;.

512

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

A primary task can issue I/O requests that are processed by the secondary task. Then the task continues processing without waiting for these I/O requests to complete. If we assumel/,~c = l/piz, then this is the average time to initiate an I/O request. In the notation of Section 10.5 we have On th e average, each secondary task generates l/p11 = 1/1-+ + (I- f)l/po. l/(1 - pz) I/O requests and thus the fraction of all I/OS that overlap with a primary task, is given by: -. f”l =

f

1

(10.153)

(1-f)+&’

bP2)

In the case of p2 = 0, we get fOl = f. In Fig. 10.39b, the central-server model with overlap is shown. Here the additional node 0 with service time 0 is introduced. Once a primary task leaves the CPU, it moves to node 0 with probability f and to I/O device i with probability (1 - f)pil. The rate at which secondary tasks are generated is equal to the throughput of primary tasks at node 0. Secondary tasks are modeled by an open class of customers where all arrivals are routed to the CPU. We will assumethat the arrival process is Poisson. The arrival rate of the open class customers X02 is set equal to the throughput of the closed class (primary tasks) at node 0. The routing of the secondary task is as described earlier. A secondary task leaving the CPU visits the I/O device labeled i with probability pi2 and returns then to the CPU with probability p2 or leaves the system with 1 - ~2, respectively, For this mixed queueing network, the stability condition is ei2Xo2 < pi2 for all i where er2 = l/(1 -ps) and ei2 = piz/(l - p2), i > 1. Table10.23 Input parametersfor the central-servermodel Model Set I II III IV V VI

rCL2

-iG

lx1

0.04 0.04 0.04 0.04 0.04 0.04

1 P4

1-15

0.04 0.04 0.04 0.10 0.20 0.40

P22

P21

=

P31

= P32

P41

= P42

0.25 0.25 0.30 0.30 0.30 0.30

P51

= P52

0.25 0.25 0.10 0.10 0.10 0.10

PO2

0.00 0.50 0.00 0.00 0.00 0.00

Six sets of central-server models with 60 models for each set were analyzed by [HeTr82]. Each set contains models with a moderate utilization of the devices as well as heavily CPU and/or I/O bound cases. Each set also contains a range of values of multiprogramming levels, K, the mean service time at the CPU, l/pi, and an overlapping factor f. A model within a set is then characterized by the triple (K, 1/,~1, f) where K = 1,3,5, l/pi = 0.002,0.01,0.02,0.10,0.2, and f = 0.1,0.25,0.5,0.75. The differences

PARALLEL

PROCESSING

513

between the sets are the I/O service times and the branching probabilities pij and ~02, respectively. Furthermore they assumefor all models that the overhead processing times are equal, l/p0 = l/pi2 = 0.0008. The mean service times of the least active I/O device is varied in an interval around the mean service time of the other I/O devices. In addition the I/O accesspattern ranges from an even distribution to a highly skewed distribution. The input parameters for each set of 60 central-server models are shown in Table 10.23. As a second example, we consider the terminal-oriented timesharing systems model shown in Fig. 10.40a. Basically it is the samemodel as the centralserver model except that it has an extra IS node (node 6) that represents a finite number of terminals. The mean service time at node 6, l/psr, can be considered as the mean thinking time. The number of terminals submitting primary tasks to the computer system is denoted by K. Each primary task first uses the CPU and then moves to I/O device i with probability pil. As soon as a primary task completes service at the I/O, it returns to the CPU with probability pl or enters the terminals with probability 1 - pl. We assumethat during the execution of a primary task a secondary task is generated which executes independently from the primary task (except for queueing effects). In particular, the secondary task can execute during the thinking time of its corresponding primary task. Routing and service times are the same as in the central-server model. The approximate model of this overlap situation is shown in Fig. 10.40b. Between the I/O devices and the terminals, node 0 (with service time 0) is inserted. Therefore the throughputs at node 0 and the terminals are equal. The secondary tasks are modeled as open job classesand arrive at the CPU. The corresponding arrival rate X02is set equal to the throughput of primary tasks at node 0. The stability condition for this model is the same as in the model before: Xo2/(pr2 ’ (1 -pz)) < 1 and X02. pi2/(,t42 . (1 - ~2)) < 1 for all i > 2.

Table 10.24 1 P61

10 10 15 15 15 15 15

---1 Pll

-

1 P12

0.010 0.015

0.002 0.010

0.020 0.020 0.030

Input parameters

1 CL,

Pi,

0.04 0.04 0.04 0.04 0.04 0.08 0.08

0.25 0.25 0.25 0.25 0.25 0.25 0.25

Pl

=

pa

0.10 0.10 0.10 0.10 0.10

0.02 0.02

for the terminal K (no.

system of terminals)

1, 5, 10, 20, 30, 40, 50,60, 70, 80, 90, 100 1, 5, 10, 20, 30, 40, 50,60, 70, 80, 90, 100 25, 50, 100 25, 50, 100 25, 50, 100 1, 5, 10, 20, 30, 40, 50 1, 5, 10, 20, 30, 40, 50

This model can be interpreted as follows: Each terminal interaction can be split into two tasks. A terminal user has to wait for completion of a primary task but not for completion of a secondary task. In [HeTr82] 47 models of this type were considered. The corresponding parameter settings for this model are described in Table 10.24. Each of the models was simulated using the IBM

514

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

Sink

Fig. IO.40 tasks (b).

Terminal

model

without

overlapped

tasks

(a) and with

overlapped

PARALLEL

Table 10.25 Comparison of DES and analytic models with balanced I/O subsystem Models

approximations

Disk

CPU

Disk

P

P

Q

s

Mean Maximum

0.003 0.026

0.006 0.071

Mean Maximum

1.2 15.9

1.5 15.4

Mean Maximum

0.003 0.020

0.005 0.054

Mean Maximum

1.0 9.2

1.1 8.7

Absolute error 0.011 0.110 Relative

Absolute error 0.009 0.073 Relative

of simulation Tab/e 10.26 Comparison models with balanced I/O subsystem

Terminal Models

approximations

Disk

CPU P

Disk P

CPU s

0.002 0.010

0.002 0.008

0.429 2.261

Absolute 0.056 0.421

error

Mean Maximum

0.3 1.4

0.4 1.5

3.0 13.4

Relative 1.4 4.5

error

Mean Maximum

s

0.044 0.457

0.056 0.710

3.2 16.2

1.3 15.5

0.072 0.670

0.034 0.317

3.1 15.1

1.0 9.0

error

2.8 25.0

and analytic

x

error

2.8 26.2

Set II

515

for the central-server

CPU

Set I

Models

PROCESSING

Terminal G

for the terminal

X

T

0.115 0.674

0.010 0.036

0.228 1.141

0.6 2.7

0.7 2.2

1.7 5.4

516

ALGORITHMS

Tab/e 10.27 server

models

FOR NON-PRODUCT-FORM

Comparison of simulation with imbalance I/O

Models

CPU P

Disk P

NETWORKS

and

2

analytic

Disk

3-5

approximations

CPU

P

Set III Mean Maximum

0.003 0.025

0.007 0.076

0.003 0.025

Mean Maximum

1.2 14.7

1.4 14.4

1.9 13.9

Mean Maximum

0.003 0.029

0.007 0.078

0.007 0.069

Mean Maximum

1.4 15.8

1.6 16.2

2.1 17.4

Mean Maximum

0.003 0.024

0.005 0.059

0.008 0.102

Mean Maximum

1.4 15.6

1.5 15.3

1.6 15.9

Mean Maximum

0.002 0.011

0.002 0.180

0.007 0.067

Mean Maximum

1.0 7.2

1.2 7.2

1.3 8.1

Disk

3-5

central

A

v

0.050 0.631

3.8 18.3

2.0 13.1

1.3 14.3

Absolute error 0.010 0.044 0.095 0.555

0.030 0.323

0.055 0.636

3.1 13.6

4.2 29.2

1.4 15.9

Absolute error 0.009 0.043 0.065 0.878

0.394 5.499

0.039 0.490

3.2 37.4

11.4 73.2

1.4 15.3

Absolute error 0.008 0.041 0.032 0.394

0.592 4.224

0.019 0.149

13.7 89.0

1.1 7.1

error

2.9 27.7

error

2.9 27.0

Relative

Set VI

2

the

0.004 0.065

Relative

Set V

CT

Absolute error 0.010 0.060 0.101 0.799 Relative

Set IV

Disk

s

for

error

3.1 28.7

Relative 4.0 28.9

error

7.4 47.9

PARALLEL

PROCESSING

517

Research Queueing package RESQ, which offers a so-called split node that splits a job into two tasks that proceed independently through the network. This node type was used to model the overlap of jobs. In Table 10.25, 10.27, and 10.26, the results of the comparison of discrete-event simulation (DES) with the approximate analytic method are shown. Both absolute errors as well as relative errors are given. For each set of models, the mean and maximum absolute and relative errors are listed for utilizations, queue lengths, throughputs, and response times (for the terminal models). As we can see from the results, the approximation is particularly accurate for estimating utilizations and throughputs. For these quantities, the relative error is about 1.3010, and the maximum relative error is 17.4%. We can also see that the estimates for the mean queue length are somewhat less accurate, the mean relative error is 4.2%. The maximum relative error is very high (up to 89%). Th ese high errors always occur in very imbalanced systems such as the central-server model sets V and VI (in these sets the overlap factor is very high). If, on the other hand, the model is better balanced (lower values of fOl), we get quite accurate approximations. In this case the mean relative error for all performance measures is 1.9%, and the maximum relative error is 2.9%. 10.6.2

Fig.

10.41

Fork-Join

Systems

Task precedence graph with (u) four parallel

tasks and (b) four tasks.

In this section we consider computer systems that are able to run programs in parallel by using the fork-join or parbegin-parend constructs. The parallel programs (jobs) consist of tasks that have to be processed in a certain order. This order can be given by a task precedence graph. In Fig. 10.41a the task precedence graph for a program consisting of four parallel tasks Y”r, T2, Ts, T4 is shown, and in Fig. 10.41b another task precedence graph for a program consisting of four tasks Tr, T’, Ts, TJ is shown. In Case b, task Ts can start after both tasks Tr and Tz are finished while task T4 can be executed in parallel with task Tr , T2, and T3. The corresponding parbegin-parend constructs look as shown in Fig.10.42.

518

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

10.6.2.1 Modeling Consider a model of a system in which a series of external parallel program execution requests arrive at an open queueing network where nodes indicate the processors and external arrivals indicate the jobs [Duda87]. It is assumed that each processor can execute exactly one special task. Thus task T! always needs to be served by processor Pi. Furthermore it is assumed that processors are always available when needed. The interarrival times of jobs are exponentially distributed with mean value l/X, and the service times of the tasks are generally distributed with mean value l/pi, i = 1,. . . , N. Figure 10.43 shows the queueing network model for jobs with the structure shown in Fig. 10.41~. Figure 10.44 shows the queueing network model of a parallel system, where jobs have the structure shown in Fig. 10.41b. The fork and join operators are shown as triangles.

fig. 10.42 Parbegin-pared construct for (a) four paralleltasks (Fig. 10.41a), parallel-sequentialtasks (Fig. 10.41b)

(b) four

The basic structure of an elementary fork-join system is shown in Fig. 10.43. The fork-join operation splits an arriving job into N tasks that arrive simultaneously at the N parallel processorsPi, . . . , PN. As soon as job i is served, it enters the join queue Qi and waits until all N tasks are done. Then the job can leave the fork-join system.

Fig, 10.43 Fork-join system.

A special caseof the elementary fork-join system is the fission-fusion system (see Fig. 10.45), where all tasks are considered identical. A job can leave the

PARALLEL

PROCESSING

519

Fig. 10.44 Queueing network model for the execution of a job with the task precedence graph shown in Fig. 10.41b.

system, as soon as any N tasks are finished. These tasks do not necessarily have to belong to the samejob.

Fig. 10.45

Fission-fusion

system.

Another special case is the split-merge system (see Fig. 10.46) where the N tasks of a job occupy all N processors. Only when all the N tasks have completed, can a new job occupy the processors. In this casethere is a queue in front of the split station and there are no processor queues. This variant corresponds to a blocking situation: A completed task blocks the processor until all other tasks are finished. Pl

PN

Fig. 10.46

Split-merge

system.

10.6.2.2 Performance Measures In this section some performance measures of fork-join systems are defined. They are based on some basic measuresof queueing systems that are defined as follows (seeFig. 10.47 and Section 7.2.1):

520

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

-Ts,l

TS,N

T9,2

(4

DTJ,N

\

(b) Fig. 10.47 System used for performance allel system.

/

TFJ

comparison:

(a) sequential

system, (b) par-

Ts,+ = The mean responsetime of task Ti belonging to a purely sequential

job Ts = The mean -response time of a purely sequential job, equal to the sum of all Ts,i FFJ = The mean responsetime of parallel-sequential jobs with fork-join synchronization TJ,~ = The mean time spent by task Ti waiting for the join synchronization in queue Qi si = The mean number of tasks waiting for the join synchronization in queue Qi TJ = The mean join time KF.J = The mean number of parallel-sequential jobs with fork-join synchronization In addition to these measures,the following other measuresare of interest for parallel systems:

PARALLEL

521

PROCESSING

Speedup: Ratio of the mean response time in the system with N sequential tasks to the mean response time in a fork-join system with N parallel tasks:

Ts GN=TFJ.

(10.154)

Normalized Speedup: G=$,

GE [+,l]s

(10.155)

Synchronization Overhead: Ratio of the mean time that tasks have to wait altogether in the join queues until all tasks are finished to the mean responsetime of the fork-join system: N

sN

c TJ,i = -?l=l %J ’

(10.156)

Normalized Synchronization Overhead: s = $,s

E [O,l].

(10.157)

Blocking Factor: The blocking factor is the average total number of blocked tasks in the join queues: BN = 2Qi.

(10.158)

i=l

Normalized Blocking Factor: The normalized blocking factor is given by: B=N.

BN

(10.159)

10.6.2.3 Analysis The method of [DuCz87] for the approximate analysis of fork-join systems is based on an application of the decomposition principle: We consider fork-join systems as open networks and replace the subnetwork that contains the fork-join construct by a composite load-dependent node. The service rates of the composite node are determined by analyzing the isolated (closed and) short-circuited subnetwork. This analysis is done for every number of tasks (1. . . 00). For practical reasons this number is, of course, limited. During the analysis, the mean number of tasks in the join queues is determined by considering the CTMC and computing the corresponding probabilities using a numerical solution method.

522

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

&I

Pl

PN

Fig, IO.48

:

Closed network

QN

r

with N- k tasks.

Consider an elementary fork-join system with N processors as shown in Fig. 10.43. To analyze this system, we consider the closed queueing network shown in Fig. 10.48. This network contains N. Ic tasks, Ic = 1,2,. . .. The analysis of the network, via the numerical solution of the underlying CTMC, gives the throughput rates X(j), which are used as the loaddependent service rates p(j) of a FES node that replaces the fork-join system (see Fig. 10.49). Th e analysis of the FES node finally gives the performance measuresof the overall system.

Fig. 10.49

Composite

service station.

In the similar fashion, the performance measuresof more complex fork-join systems can be computed because a job can have task precedence graphs with several nested fork and join constructs. The model for the service of a job has then the form of a series-parallel network with subnetworks that again contain fork-join systems. In Fig. 10.44 we have already given an example of such a model. The described decomposition principle is applied in this case several times in order to solve the fork-join subnetworks numerically so that they can be replaced by FES service stations. Example 10.16 Consider a two-server fork-join queue. Here we assume that the service times at both servers are exponentially distributed with mean l/p. This system is shown in Fig. 10.43 (with N = 2). In order to solve the fork-join queue, we consider the FES queue with state-dependent service rate, shown in Fig. 10.49. The decomposition step consists of analyzing the closed subsystem with 21ctasks, k = 1,2,. . . (Fig. 10.48). The service rate of the FES in Fig. 10.49 is set equal to the throughput of the closed subsystem. The solution of the FES gives overall performance measures. At the level of the closed subsystem we deal with forked tasks and delays caused by the join primitive, while at the FES level we obtain the performance measuresfor jobs.

PARALLEL

523

PROCESSING

Let X(t) denote the total number of tasks in collectively both the join queues at time t. The stochastic process is a CTMC with the state space {i = 1,2,.

. . ) k}.

The state diagrams of the CTMCs for various values of Ic are shown in Fig. 10.50. Each of these CTMCs is a finite birth-death process, and thus the

Fig. IO.50

CTMC

for the total number

of tasks waiting

in the join queues.

steady-state probabilities can be derived using Eqs. (3.11) and (3.12): 1 ro= 1+21c’ 7ri = 27r0, i = 1,2,. . . ) It,

(10.160)

k = 1,2 )....

In order to calculate the throughput of jobs A( Ic), we observe that a successfuljoin corresponds to a transition at rate ,Qfrom state i to state (i - 1) in the CTMCs of Fig. 10.50. Thus assigning reward rate p to each of the states i = 1,2,. . . ) Ic, we obtain the needed throughput as the expected reward rate in the steady states.

X(lc) =/&i = &,

k = 1,2,.

..*

(10.161)

i=l

State-dependent service rates of the FES are then set equal to the throughput

of the closed subsystem: p(k) = X(k),

k = 1,2,. . .. The solution of the FES

queue then gives the steady-state probabilities of the number of jobs in the

524

fork-join

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

queue and is given by:

Vk = vOpki=lk

)

II= 1,2 )...)

n 2i i=l

and with Eq. (3.12): 1

(10.162)

v(-J =

fi2i+1

l+

E Pk i=l

k=l

fJ 2i i=l

=

(1 -

t$7

(see [BrSeSI-37P.32),

where p = X/p. These results hold for p < 1, which is just the ergodicity condition for fork-join queues. Knowing the probabilities of Eq. (10.162), the expected number of jobs in the system is:

j&=-&,k=3.P2

l-p'

k=l

(10.163)

and, using Little’s theorem: T&J =

KFJ 3 1 - x = 5 * /V(l - p)’

(10.164)

If these values are compared to the corresponding ones for the M/M/l queue, it can be seen that the mean number of jobs and the mean response time of the fork-join queue are both 3/2 times larger than those of the M/M/l queue. The performance indices of the synchronization primitives are derived in the following: With the mean responsetime Ts,i of task i of a purely sequential job (= mean responsetime of a M/M/l queue, Eq. (6.15))

Ts,+ = ~ 41

1

i = 1,2,

- P> ’

we obtain the mean responsetime of a purely sequential job:

T,=&,,p i=l

2 PO -P)'

Equations (10.154) and (10.164) are used to derive the speedup:

Ts GN=TFJ=3.

4

PARALLEL

The normalized

speedup is computed

using Eq. (10.155).

PROCESSING

525

With N = 2 we get:

To compute the synchronization overhead SN, we need the average time a task spends in the join queues. This time is just given by the difference between ?;FJ and TM/M/r (where TM/M/r is the time a task spends in the servers). If:

then the synchronized

and the normalized

overhead is given by (Eq. (10.156)):

synchronization

overhead (Eq. (10.157)):

In order to obtain the blocking factor BN, we need the mean number of tasks in the join queues. This number can be determined from Eq. (10.165), using Little’s theorem:

and the normalized

blocking

factor is given by:

Note that the speedup as well as the synchronization overhead do not depend on the utilization p, and that the speedup is rather poor. It can be shown that the speedup GZ increases from 4/3 to 2 if the coefficient of variation of the service time decreases from 1 (exponential distribution) to 0 (deterministic distribution). In Tables 10.28, 10.29, and 10.30, the speedup is given for different distribution functions of the service times. It is assumed that jobs arrive from outside with arrival rate X = 0.1. The service rate is ,Q = 10. For the case cx < 1, we use an Erlang-lc distribution and get the results shown in Table 10.28. For cx > 1, we use a hyperexponential distribution and get the results shown in Table 10.29. Because of the phase-type service time distribution, the method discussed so far can easily be extended. Underlying

526

ALGORITHMS

Tab/e

10.28

FOR NON-PRODUCT-FORM

NETWORKS

System speedup with Erlang-k

OeO

service times

Erlang-k

Exact var(X)

distributed

ii+5

iA

cx

0.0

&

53

Speedup

2.0

1.760

1.671

1

1

1

1

1

600

500

400

300

200

1 100

5

5

-A

5

5

1.0

1.630

1.605

1.569

1.524

1.453

1.333

Table 10.29 System speedup with hyperexponentially service times . var(X) Speedup

distributed

2.0

4.0

6.0

8.0

10.0

20.0

1.060

1.043

1.029

1.028

1.027

1.026

CTMCs will now be more complex than those shown in Fig. 10.50. We chose to generate and solve these CTMCs using the MOSES tool (see Chapter 12). Finally we show in Table 10.30 the system speedup if the service times are normally distributed with standard deviation 0~ and mean value p = 10. The results in this case were obtained using DES. Using a similar approach, Tab/e

10.30

System speedup with normally

distributed

service times

var(X)

0.001

0.005

0.01

0.02

0.08

0.1

0.2

0.4

Speedup

1.697

1.525

1.487

1.463

1.439

1.437

1.433

1.431

performance measures for the fission fusion system (see Fig. 10.45) and the split merge system (see Fig. 10.46) can be computed [DuCz87]. For fork-join systems with a large number of processors, [Duda87] developed an approximate method for the analysis of the subnetworks. To demonstrate this method we consider a series-parallel closed fork-join network as shown in Fig. 10.51. In this method it is assumed that all service times of the tasks are exponentially distributed. The method is based on construction of a product-form network with the same topology as the original fork-join subnetwork and approximately the same number of states in the corresponding CTMC. The main difficulty when following this procedure is to determine the number of states in the fork-join subnetwork. If N denotes the number of parallel branches where the ith branch contains exactly ni nodes and Ic tasks, then the number of ways to distribute Ic tasks to ni + 1 nodes (including the join queue &i) is given by: (10.166)

PARALLEL

PROCESSING

527

Nk Tasks Fig.

10.51

Series-parallel

closed fork-join

network.

Then the number of ways to distribute N. Ic tasks to all nodes is given by the following expression:

fi

(10.167)

xi 0k .

i=l

This number can be limited further. If there is exactly one job in each join queue Qi, then the jobs are passed immediately to the first station. Thus j-J:, zi(k - 1) of th e given combinations are not possible because this is the number of possible ways to distribute Ic - 1 tasks over ni stations, given that there is one task in each join queue Qi . The number of states in the fork-join subnetwork is therefore given by: &J(k)

= f&i(k) i=l

- fiZi(k

- 1).

(10.168)

i=l

In the following steps, the algorithm for the analysis of fork-join networks with possibly several nested fork-join subnetworks is given. The replacement of these fork-join subsystems by FES nodes is done iteratively. Choose one subsystem that does not contain any other fork-join structures and construct a closed network with N. Ic tasks by using a short circuit. Compute the number of states .&J(IC) of this closed subnetwork for k = 1,2, . . . (Eq. (10.168)).

528

ALGORITHMS

FOR NON-PRODUCT-FORM

Compute

the throughput

NETWORKS

of the subnetwork.

-1 Construct a product-form network with exactly the same M = CL, ni number of nodes as the subnetwork. The number of jobs K in the product-form network is chosen so that the number of states in the productform network is approximately equal to the number of states in the fork-join subnetwork. Since the state space structures of both networks are almost identical, the throughput of the product-form network XPF(K) will be approximatively equal to the throughput of the fork-join network XFJ(IC). Thus we need to determine K by using Eqs. (10.166) and (10.168) so that for Ic = 1,2,. . .: K

with &F(K)

: lzFJ(k)

-

zPF(K)I

= .z~-1(K)

=

mp[ZFJ(k)

-

ZPF@)i

?

(10.169)

holding.

1-1 Solve the product-form network for all possible numbers of jobs K using an algorithm such as MVA and determine the throughput as a function of K. This throughput XPF(K) of the product-form network is approxXFJ(/C) of the fork-join subnetwork for the imately equal to the throughput value of K corresponding to Ic. Replace the subnetwork in the given system by a composite node whose load-dependent service rates are equal to the determined throughput rates: p(k)

network.

=

XFJ(IC)

for k = 1,2.. . .

If the network under consideration contains only one FES node, nalysis of this node gives the performance measures for the whole Otherwise go back to Step 1.

From these performance measures, special characteristic parallel systems such as the speedup GN and the synchronization SN can be computed. The algorithm

is now demonstrated

Example 10.17 Consider depicted in Fig. 10.41b. The network is presented in Fig. service times of the tasks are 1 - = 10, CL1

values of overhead

on a simple example.

a parallel program with the precedence graph model in the form of a series parallel fork-join 10.44. The arrival rate of jobs is X = 0.04. All exponentially distributed with the mean values: -

1

= 5,

CL2

-

1

CL3

= 2,

-

1

ZE 10.

l-b

The analysis of the model is carried out in the steps that follow: At first the inner fork-join and P2, is chosen and short-circuited. 2. k tasks, k = 1,2, . . . (Fig. 10.52).

system, consisting of the processors Pi This short circuited model consists of

PARALLEL

fig. 10.52 Table 10.31

Short-circuited

Number

PROCESSING

529

model.

of states in the subsystem.

k

1

2

3

4

5

6

7

8

9

10

&‘J K m&F

3 3 3

5 5 5

7 7 7

9 9 9

11 11 11

13 13 13

15 15 15

17 17 17

19 19 19

21 21 21

The number of states ZFJ of this subsystem is determined using Eq. (10.168). For Ic = 1,. . . , 10, the results are shown in Table 10.31. Determine the throughput of the subnetwork. -1 Construct a product-form network with N = 2 nodes. The number of jobs K in this network are determined using Eq. (10.169). For k = 1,2..., 10 the values for K and the number of states ZPF(K) are given in Table 10.31. -1 Using MVA we solve the product-form network, constructed in Step 3.1, for all possible number of jobs K. The throughputs are given by: X43)

= 0.093,

&F(5) = 0.098,

&F(7)

= 0.100,

&(g)

= o-100,

These values approximately correspond to the throughputs XFJ(~) of the forkjoin subnetwork: XFJ(~) = XPF(3) = o-093, XFJ(2) = &F(5) = [email protected], &J(3) = XPF(7) = 0.100, XFJ(4) = XPF(g) = 0.100,

The analyzed fork-join subnetwork, consisting of the processorsPI and P2, is replaced by an FES node (see Fig. 10.53) with the load-dependent service rates p(k) = XFJ(k) for k = 1,2, . . . .

530

ALGORITHMS

FOR NON-PRODUCT-FORM

fig. 10.53

Fork-join

NETWORKS

subnetwork

with one FES node.

This network contains more than one node and therefore we return to Step 1.

fig. 10.54

Short-circuited

network.

The short circuit of the network shown in Fig. 10.53 results in a closed network with 2. k tasks, k = 1,2,. . . (Fig. 10.54). The possible number of states ZFJ in this network is given by Eq. (10.168). For k = 1,. . . , 10 the values are shown in the Table 10.32. Tab/e 10.32

The possible number of states

ZFJ

as a function

of k = 1,. . . , 10

k

1

2

3

4

5

6

7

8

9

10

ZFJ

5 2 6

12 3 10

22 5 21

35 7 36

51 9 55

70 10 66

92 12 91

117 14 120

145 16 153

176 17 171

K ZPF

Determine the throughputs. 1m3.1) Construct a product-form network consisting of N = 3 nodes and determine the number of jobs K in the network using Eq. (10.169). For k = l,... , 10 the values of K and .&F(K) are given in Table 10.32. \-I Solve the network for all number of jobs K. For the computed throughputs, the following relation approximately holds: XpF (K) = XFJ( k).

PARALLEL

PROCESSING

531

ITsing the MVA we get:

~~~(1) = X42) &~(3)

= 0.062,

&J(Z)

= APF(5) = 0.082,

= XPF(3) = 0.072,

XFJ (4) = xpF( 7) = 0.087,

XFJ(5) = &(9)

= 0.089,

XFJ (6) f8 iiPF(10) = o-090,

XFJ(7) = XPF(l2)

= o-092,

XFJ(~) % XPF(14) = o-093,

XFJ(S)

x

XFJ(~O) z XpF(l7)

Xp~(16)=0.094,

= 0.094.

The fork-join subnetwork is now an FES node (see Fig. 10.55) with the load-dependent service rates $(Ic) = XFJ(~), k = 1,2,. . . .

Fig.

10.55

FES queue for Example

10.17.

The fork-join network is now reduced to only one node and therefore the iteration stops and the performance measuresof the system can be determined by analyzing the FES node of Fig. 10.55. For this analysis we determine at first the steady-state probabilities of the node using Eqs. (3.11) and (3.12): r. = 0.428,

xl = 0.287, 7r2= 0.155,

7r3= 0.076,

7r4

= 0.034,

7r6= 0.007,

7r7

= 0.003, r8 = 0.001,

TT5

= 0.015,

7rg= 0.001, n-10= 0.000. These probabilities are then used to determine the mean number of jobs in the fork-join model, Eq. (6.8): KFJ

= 1.116,

and the mean response time, Eq. (6.9): TFJ

=

27.89.

With Eq. (6.15) and:

r;,=&T,;+=f: i=l

i=l

l /-a

we obtain: ?& = 41.76.

-

Pi>

’

532

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

To obtain the speedup, we use Eq. (10.154) and get: GN = 1.497. From this result we can compute the normalized GN

G = N

speedup (Eq. (10.155)):

= 0.374.

To obtain the synchronization overhead SN, we need the mean waiting time queues. First we consider the fork-join TJ,.i, (i = 1,2,3,4) in synchronization queue of node 1 and node 2:

TJ& = TFJ12 - T.i,

(10.170)

where TF.J~ 2 is the mean response time in the fork-join construct consisting of node 1 and node 2. It is obtained from the state-dependent service rate ,~(lc) for the composite node of the inner fork-join construct. At first we need the state probabilities, using Eqs. (3.11) and (3.12): x0 = 0.580,

7rl = 0.249,

r2 = 0.102,

= 0.041,

r4 = 0.016,

7r5

7r3

r6 = 0.003,

= 0.001,

7r7

= 0.007,

iTTf.3 = 0.000,

and the mean number of jobs: &J12

= 2

hrr, = 0.701.

k=l

If we apply Little’s theorem, we obtain for the mean response parallel-sequential job with fork-join synchronizations: K

T FJu = -

FJn x

time of a

= 17.514.

In order to obtain the mean response time, we apply Eq. (6.15) and get: 1

= Pl(1

-

= 16.667,

T&=6.25.

Pl)

Using Eq. (10.170) we finally get the the results: TJ,1

=

- T1 = 0.874,

%J12

TJ,2

=

11.264.

From Fig. 10.53 we have: TJ,~

=

TFJ

-TFJ~~

-T3,

TJ,4

=

TFJ

-574.

NETWORKS

Applying

WITH

ASYMMETRIC

NODES

533

Eq. (6.15) again: T3 = 2.174,

T4 = 16.667.

Together with TFJ and TFJ~~ from the preceding we get: TJ,3 = 8.202,

TJ,4 = 11.233,

which finally gives us the synchronization 6 s

The normalized

N

synchronization

=i=l

-

overhead (see Eq. (10.156)):

TJ,i

= 1.132.

%J

overhead is given by (Eq. (10.157)): SN s=N = 0.292.

Now it is easy to calculate the blocking obtain, via Eq. (10.158):

BN = k?Ji

factor.

Using Little’s

theorem,

we

= 1.263,

i=l

and with Eq. (10.159) the normalized

blocking factor is given by:

B = 1BN

4

= 0.316.

A comparison of the mean response time computed with the preceding approximation with DES shows that the differences between the DES results (?T = 26.27) and the approximation results are small in our example. For fork-join systems with higher arrival rates, the differences become larger. It can be shown in general that the power of the fork-join operations is reduced considerably by join synchronization operations. Also the speedup is reduced considerably by using synchronization operations. Other approaches to the analysis of parallel programs can be found in [Flei89], [MiiRo87], [NeTa88], [SaTr87], and [ThBa86]. Problem 10.5 Consider a parallel program with the precedence graph shown in Fig. 10.56. The arrival rate is X = 1 and the service rates of the tasks are ~1 = 2, ~2 = 4, ,Q = 4, ,Q = 5. All service times are exponentially distributed. Determine the mean response time and the speedup.

10.7

NETWORKS

WITH

ASYMMETRIC

Whenever we have considered multiserver assumed that all the servers

NODES

nodes in a network, we have always

at the node are statistically

identical.

We now

534

ALGORITHMS

FOR

fig.

NON-PRODUCT-FORM

NETWORKS

Precedence graph for Problem

10.56

consider the case of networks where servers ent service times. We first consider closed followed by open networks with asymmetric cases we will only obtain an approximative 10.7.1

Closed

10.5.

at a multiserver node have differnetworks with asymmetric nodes nodes. Note that in each of these solution.

Networks

In this section we show how to modify the summation method, the mean value analysis, and the SCAT algorithm to deal with asymmetric nodes[BoRo94] as introduced in Section 9.1.2. 10.7.1.1 Asymmetric SUM (ASUM) To extend the SUM (seeSection 9.2) to queueing networks with asymmetric nodes, we need to adapt the formula for computing the mean number of jobs (see Eqs. (9.15) and (6.28)): Ki

= mi

Pi

f pi +

1-K-me--l K-m,

*pm,,

Type-l,

‘Pi

with:

‘mi(Pi)

=

m;! . (1 - pi) me1 (mi * pi)” + (mi . pi)mi ’ k! k=O mi * (1 - pi)

For symmetric nodes, pi is given by: pi =

Ai ____ 3 mi * pi

and for asymmetric nodes we can use Eq. (6.136) or Eqs. (6.150) and (6.146), respectively. Since the SUM is an approximate method, so is the extended SUM. By replacing p; and Pm, by the corresponding formulae, the BOTT can be extended to queueing networks with asymmetric nodes in the same way.

NETWORKS

WITH

ASYMMETRIC

535

NODES

10.7.1.2 Asymmetric MVA (AM VA) The MVA was developed by Reiser and Lavenberg [ReLa80] for the exact analysis of product-form networks and is based on Little’s theorem and the arrival theorem of Reiser, Lavenberg, Sevcik, and Mitrani (see Section 8.2) [LittBl, ReLa80, SeMi81]. To extend the MVA to networks with asymmetric nodes, the equation for the calculation of the mean response time for symmetric -/M/m nodes:

a.

i

i

1 + Ki(Ic - 1) + c (Mi - j - 1) *%(j I k - 1) ? j=o ) (10.171)

needs to be extended. Here 7ri(j 1Ic) is the probability that there are j jobs at the ith node, given that there are Ic jobs in the network. This condition probability ri(j 1Ic) is given by:

*dj-w-l),

j=l,...,

mi-1,

(10.172)

and ri(O 1Ic) is obtained by:

;Iri(O )Ic)= 1- L

mi

2 *X(k)+ mel(mi - j) *7ri(j) Ic) * j=l [ pi I

(10.173)

In Eq. (10.173), the factor ,Q,. rni is the overall service rate of the symmetric -/M/m node. For asymmetric nodes m; . pi is replaced by:

Thus the mean response time of an asymmetric node can be computed as follows:

T@) = -A2 j=l

(

m,-2

l+IiTi(k-l)+

,Uij

C j=O

( mi-j-l)*ni(jIIc-1)

. ) (10.174)

Because of the different service rates, it is necessary to take into account which servers are occupied as well as the number of servers when calculating

the marginal probabilities

n&j 1 k). If we assume that the free server with the

highest service rate is selected and the service rates are arranged in descending

ALGORITHMS

536

FOR NON-PRODUCT-FORM

order, then the marginal

NETWORKS

probabilities

are given by: (10.175)

m,-1

7ri(O 1 k) = 1 - *

* X(k) - -&- * C( mi j=l c Pi1

rni - j) .7ri(j

1 k).

(10.176)

l=l

If we consider Eq. (10.173) in more detail, we see that the factor: ei . X(k) E

Pi1

l=l

is the utilization pi and, therefore, Eq. (10.176) can be rewritten as: ni(O

1k) = 1 - pi(k) - k

z

* mgl(mi

- j) * ni(j I k).

j=l

Asymmetric SCAT (ASCAT) The core of the SCAT algorithm is a single step of the MVA in which the formula for the mean response time, Eq. (10.174), is approximated so that it depends only on K and not on (K- 1). However, it is necessary to estimate the marginal probabilities ni(j 1K - 1) that need a lot of computation time. For this reason, another formula for the mean response time is derived from Eq. (6.29). This derivation depends only on ??i and pi. For -/M/l nodes we apply Little’s theorem to the following equation: 10.7.1.3

li’,

=

pi

1 - pi’ and after reorganizing we get: Ti

-i(l

=

+Ki),

Pi

which is the basic equation of the core of the SCAT algorithm for -/M/l nodes. Similarly, from Eq. (6.29) we obtain:

and for -/M/m

nodes we have: Ti =

&

* (mi 2’

2

+

Ki

+

pmi(pi)

-

mi

* pi).

NETWORKS

WITH

This formula can easily be extended to asymmetric

ASYMMETRIC

NODES

537

nodes as follows:

1

Ti=-.

Crni + Ki + Pm% (pi) - mi . pi) . 2

Pij

j=l

At this point we proceed analogously to the SUM by using the equations belonging to the selected method to calculate pi and Pm%(pi). Our experience with these methods suggeststhat the mean deviation for ASUM and AMVA is in the order of 2-3% for the random selection of a server as well as for the selection of the fastest free server. For ASCAT, the situation is somewhat different. For random selection the deviation is considerably high (7%), while for the selection of the fastest free server the mean deviation is only of the order of 1%. Therefore, this method can be recommended. ASUM, AMVA, and ASCAT can easily be extended to multiclass queueing networks [AbBo97]. 10.7.2

Open

Networks

Jackson’s method for analyzing open queueing networks with symmetric nodes (see Section 7.3.4) can also be extended to analyze open queueing networks with asymmetric nodes. Note that this will result in an approximate solution. Jackson’s theorem can be applied to open single classqueueing networks with symmetric -/M/m-FCFS nodes. The analysis of an open Jackson network is done in two steps: Calculate the arrival rates Xi using traffic Eq. (7.1) for all nodes i = l...N. Consider each node i as an elementary -/M/m-FCFS queueing system. Check whether the ergodicity (p < 1) is fulfilled and calculate the performance measuresusing the known formulae. Step 1 remains unchanged when applied to asymmetric open networks. In Step 2, formulae given in Section 6.16 for asymmetric -/M/m-FCFS queueing networks should be used instead of formulae for symmetric -/M/m-FCFS queueing systems. Similarly, the BCMP theorem for open networks (see Section 7.3.6) can be extended to the approximate analysis of asymmetric queueing networks. Using the closing method (see Section 10.1.5) AMVA, ASCAT and ASUM can also be applied to open asymmetric queueing networks. Furthermore, it is also possible to apply the closing method to open and mixed queueing networks with asymmetric nodes. Example 10.18 To demonstrate different methods for networks with asymmetric nodes we use the example network shown in Fig. 10.57. Node 1 is an asymmetric -/M/2, nodes 2 and 3 are of -/M/l type, and node 4 is -/G/co. The routing probabilities are: 1312 = 0.5,

p13

= 0.5,

p24 =p34

=p41

= 1,

538

ALGORITHMS

FOR NON-PRODUCT-FORM

fig. 10.57

NETWORKS

Closed network with one asymmetric

node.

and the service rates are: /All = 3.333,

/Liz = 0.666,

/.L~= 1.666,

/.L~= 1.25,

ru4= 1.

The total number of jobs in the network is K = 3. From the routing probabilities we obtain the following visit ratios: el = 1,

e2 = 0.5,

e3 = 0.5,

e4 = 1.

We start with the ASUM method, which consistsof the samesteps asthe SUM algorithm (see Section 9.2), but with different formulae for the utilization pr and the probability of waiting Pml. For the stopping condition we use E= 0.001. Initialization:

Xl = 0 and X, = min i

=Q.

Bisection: r---T-] STEP 2 i

h + AL X = 2=

1.25.

-1 Calculation of the values fi(Xi), i = 1,. . . ,4. In order to get the result for node 1, Eqs. (6.146) and (6.150) for the exact analysis of networks with asymmetric nodes are used (selection of the fastest free server). With: p1 =

e1 Pll +

1.25 z---z 4

*A p12

0.3125

and: pm1 2-e

l-c

-= PA”’ N

1

1 - 0.3125

* -0.111 = 0.153, 1.050

we get:

Jwl)

= 2Pl + PI * Pm1 = 0.673.

NETWORKS

Accordingly

WITH

ASYMMETRIC

NODES

539

we obtain:

IT,(X,) = -E1 K3(X3)

=

+ x4

K,(X,)

= -

= 0.500

with

p2 = 0.375,

= 0.750

with

p3 = 0.500,

iP2

3P3

= 1.250,

P4

which yields: g(X) = -&xi)

= 3.173.

i=l

-1 X = 1.25.

Check the stopping condition. Becauseof g(X) > K, we set X, =

r] STEP 2 i

X= v

-1

Calculation of the values fi(&), 171

(h)

=

2Pl

+

= 0.625.

p1

= -Jk-

K&42)

1 -

= 0.319. with p1 = 0.156 and Pm, = 0.0386,

= 0.214 with p2 = 0.188,

$P2

= -&-

E3(X3)

* en,

i = 1,. . . ,4. With:

l-

= 0.300

with p3 = 0.250,

$P3

x4

77,(X,)

= Lq = 0.625,

g(X) = &Ti(hi) i=l (-2.31

= 1.458.

Check the stopping condition. Becauseof g(X) < K, we set Xl =

X = 0.625.

STEP 2 (r---T-j

I

-2.2 =

s?;,(X,)

=A

?74(X4)

= 0.938.

Calculation of the values fi (A,), i = 1, . . . ,4. With

K(h)

E3(X3)

*

X=

=

=

2p1

+

p1

= 0.489. with p1 = 0.234 and Pm, = 0.0881,

= 0.346

1 -

$P2

dz1 -

$P3

x4 L

* En,

with p2 = 0.281,

= 0.500 with p3 = 0.375, =

0.938,

ALGORITHMS

540

FOR NON-PRODUCT-FORM

NETWORKS

we get: g(X) = &?i(hi)

= 2.273.

i=l

-1 x = 0.9375.

After fulfilled.

Check the stopping

condition.

Because of g(X) < K, we set Xl =

11 iterations we have g(X) = 3.000 and the stopping condition For the overall throughput we obtain X = 1.193, which yields: x1

=

1.193,

For the utilizations p11 =

0.294,

X2

0.596,

=

X3

0.596,

=

X4

=

1.193.

of the nodes we get: ~12

= 0.317,

pl = 0.298,

p2 = 0.358,

p3 = 0.477,

and for the mean number of jobs in a node: K1 = 0.638,

K2 = 0.470,

Now we analyze the network

?T, = 0.700,

using the AMVA

??, = 1.193.

algorithm.

Initialization: K(O)

= F2(0)

Iteration k=

over the number

Pi(l

IO) = 0.

of jobs in the network

1.

I-2.11

Mean response times: Ti(1)

= Pll

+ I-L12

(1 -t%(O)

+m(0

T2(1)

= ;

(1+17,(o))

= 0.6,

T3(1)

=

;

(1+

=

;

= 1.

?54(1)=

I-2.2

p1(0 ( 0) = 1,

= 0,

= K3(0)

K3(0))

0.8,

Throughput: X(l)=

l 5 i=l

e;Ti(l)

= 0.455.

IO)) = 0.5,

starting

with

is

NETWORKS

1-1

NODES

Mean number of jobs: xi(l)

= A(l)rr(l)er

KS(~) = X(l)??s(l)es Iteration

WITH ASYMMETRIC

= 0.227,

x2(1)

= X(l)T2(l)e2

= 0.136,

= 0.182,

F4(1)

= X(l)TJ(l)e,

= 0.455.

for k = 2:

I]

Mean response times: Fi(2)

=

IL CL11 +

p1(0 1 1) = 1 -

(l-tJG(l)

elx(l) Pull

- -&(l

1 1) = 0.818

= ;

(1 +x2(l))

= 0.682,

Ts(2)

= ;

(1 +Ka(1))

= 0.945,

i

and

10) = 0.136,

%(2)

T4(2),=

1 1)) = 0.511,

+ Pl2

elx(l) p1(1 1 1) = -pl(O

I]

+p1(0

p12

= I.

Throughput: X(2) =

2 e eiTi(2)

= 0.860.

i=l

[]

Iteration ]STEP]

Mean number of jobs: 171(2) = X(2)&(2)el

= 0.440,

K2(2)

= X(2)T2(2)e2

= 0.293,

X3(2)

= 0.407,

K4(2)

= X(2)574(2)e*

= 0.860.

= X(2)T3(2)e3

for k = 3: Mean response times: Tr(3)

=

l Pll

+

(1 + K,(2) P12

+ p1(0 ) a>) = 0.530,

541

542

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

where: p1(0 1 2) = 1 -

+

1 12) = 0.679

and

I-412

pr(1 12) = ~Pl(O e1x(2)

-2.2

- --&(

e1x(2) Pll

I 1)

=

0.211,

T2(3) = i

(1 +K2(2))

= 0.776,

T3(3) = ;

(l+&(2))

= 1.125,

Throughput: X(3) =

4

= 1.209.

3

C eiTi(3) i=l

(1

Mean number of jobs: Fr(3)

= X(3)T1(3)el

= 0.641,

K2(3)

= X(3)T2(3)e2

= 0.469,

Z,(S)

= X(3)T3(3)e3

= 0.681,

K4(3)

= X(3)T4(3)ed

= 1.209.

The algorithm

stops after three iteration

X1 = 1.209, For the utilizations

X2 = 0.605,

steps. For the throughput

X3 = 0.605,

we get:

X4 = 1.209.

of the nodes we obtain: p1 = 0.302,

p2 = 0.363,

p3 = 0.484,

and for the mean number of jobs in a node: K1 = 0.641,

K2 = 0.469,

K3 = 0.681,

r4 = 1.209.

Finally, the analysis of the network is performed using the ASCAT algorithm, where we use the approximate Eqs. (6.135) and (6.136) and Eqs. (6.27) and (6.28) f or asymmetric multiple server nodes. Modified input parameters:

core algorithm

Ki(K) = $

=m

for K = 3 jobs in the network

and Di(K)

=0

for i = 1,...,4.

with

the

NETWORKS

-1

Estimate

WITH

ASYMMETRIC

NODES

values for the ?Ti(K - 1) from Eq. (9.5): xi(s)

Ki(2)

= 23

= 0.5

for i = 1,. . . ,4.

For the next step we need an initial value for the throughput: +?)

-1

=

P4

%(2)

= 0.5.

One step of MVA: Tr(3)

1

= Pll

+

P12

*

(ml + %(2)

+ &nl (PI) - mlpl)

= 0.569,

with: elA(2) Pl

cl11

%(3)

=-

=

0.5 4

+cL12

= 0.125

= ;

(1 +17,(2))

= 0.9,

TS(3) = ;

(1+ Ks(2))

= l.2,

T*(3) = i

= 1.

and Pm1 (pr) = 0.0278,

Throughput: X(3) =

4 C i=l

3

= 1.145.

eiTi(3)

Mean number of jobs: Kr(3)

= X(3)T1(3)el

= 0.652,

??p(3)= A(3)T2(3)e2

= 0.515,

?T3(3) = X(3)T3(3)es

= 0.687,

K,(S)

= 1.145.

[STEP]

Check the stopping /$l’(3) max i

-1

{

Estimate J?,(2)

= 0.435,

= X(3)T4(3)e4

condition: - $O’(3) 3

1 = 0.132 > E = 0.01. 1

values for the Ici(K - 1) from Eq. (9.5): li;2(2) = 0.344,

Ks(2)

= 0.458,

z4(2)

For the next step we need an initial value for the throughput: x(2) = &q(2)

= 0.764.

= 0.764.

543

544

ALGORITHMS

r-1

After jobs:

FOR NON-PRODUCT-FORM

NETWORKS

One step of MVA:

three iterations

zr(3)

we get the following

= 0.626,

Ka(3)

= 0.475,

Modified core algorithm the input parameters: Ki(2)

-1

= ;

Estimate

= 0.5

Ka(3)

for the mean number

= 0.702,

Ed(S)

and Di(2)=0

= 27m4

for i=1,...,4.

from Eq. (9.5):

= 0.

for i = 1,...,4.

For the next step we need an initial value for the throughput: X(l)

[ml

=

err,(l)

p4

= 0.25.

One step of MVA: Tr(2) =

IL CL11 +

* cm,

Pl2

+

K(1)

+

en1

(Pl)

-

mlpl)

=

0.533,

with: Pl

fd(l>

= Pull

T2(2)

= i

T3(2) = ; ?&)

= i

+

= 7

= 0.0625 and Pm1(pl) = 0.00735,

Pl2

(l+&(1))

=0.75,

(l+K3(1))

= Lo,

= 1.

Throughput: X(2) =

4

2

C eiTi(2) i=l

= 0.831.

of

= 1.198.

for (K - 1) = 2 jobs in the network

values for the ?ri(l)

J?,(l)

results

with

NETWORKS

WITH

ASYMMETRIC

NODES

545

Mean number of jobs: Kr(2)

= X(2)T1(2)el

= 0.443,

1?3(2) = X(2)T3(2)e3

= 0.415,

-1

Check the stopping (R!“(2)

-1 X1(1)

- $O’(2) 2

Estimate

values for the xi(l) &(l)

= 0.311, = 0.831.

condition:

max i

= 0.221,

K2(2)= X(2)T2(2)e2 K*(2) = A(2)T4(2)e4

( = 0.165 > E = 0.01.

= 0.156,

from Eq. (9.5): Es(l)

= 0.208,

x4(1)

= 0.415.

For the next step we need an initial value for the throughput: X(1) = /J*K4(1) [I

= 0.415.

One step of MVA:

After jobs:

three iterations Kr(2)

= 0.434,

we get the following Kz(2)

Estimate

= 0.295,

results

I&(2)

the I?i and Di values with

for the mean number

of

1?4(2) = 0.857.

= 0.414,

Eqs. (9.3) and (9.4), respec-

t ively : Fl(3)

x1 (3)

= --j--

= 0.209,

h(3) = 3% (3) = 0.158, K3

F3(3)

= --Fj-

F4(3)

=

K4(3) --j-

(3)

JG(2) Fr(2) = 2 = 0.217, m2) F2(2) = 2 -

= 0 .148,

= 0.234, F3(2) = F =

0.399,

= 0.207,

F4(2) = 23w)

= 0 .429 .

Therefore we get: D1(3) = Fl(2) - Fl(3) = 0.00835, D3(3) = I;;(2) - F3(3) = -0.0270,

&(3)

= F2(2) - &(3) = -0.0106,

D4(3) = F4(2)

-

F4(3)

= 0.0292.

Modified core algorithm for K = 3 jobs in the network with the computed E(i (K) values from Step 1 and the Di (K) values from Step 3 as inputs.

546

ALGORITHMS

[I

FOR NON-PRODUCT-FORM

Estimate Er(2)

NETWORKS

values for the K,(K K(3) -yj-

= 2

17,(2) = 0.295,

- 1) from Eq. (9.5):

+ Dr (3)

= 0.434, > ?73(2) = 0.414, K4(2)

= 0.857.

For the next step we need an initial value for the throughput: x(2) = ~4 . F4(2) j]

= 0.857.

One step of MVA: T1(3) =

l Pll

+

p12

*

(ml + %(2)

+ pm, (~1) - mlpl)

= 0.520,

with: pr = 0.24 &(3)

= ;

and Pm1 (pr) = 0.0756, (1 + K2(2))

= 0.777,

?i3(3) = ;

(1 +Ks(2))

= 1.131,

T4(3)

= I.

=

;

Throughput: X(3) =

4

3

= 1.212.

C &T(3) i=l

Mean number of jobs: Kr(3) = X(3)T1(3)el Es(S) -1

= X(3)T3(3)es

= 0.631,

?72(3)= X(3)??2(3)e2 = 0.471,

= 0.686,

X4(3)

= X(3)T4(3)e4 = 1.212.

Check the stopping condition: /f7j1’(3) - r?i0’(3) / max i

{

3

= 0.00531 < E = 0.01. 1

The stopping condition is fulfilled. Therefore the algorithm stops and we get the following results: Throughputs: Xl = 1.212,

x2 = 0.606,

x3 = 0.606,

x4

= 1.212.

NETWORKS

WITH

BLOCKING

547

Mean number of jobs: -K1 = 0.631,

IT, = 0.471,

K3 = 0.686,

774 = 1.212.

Utilizations: p1 = 0.303,

p2 = 0.364,

p3 = 0.485.

Table 10.33 Throughput A, and mean number of jobs Ei for different methods compared with the DES results 1

2

3

4

ASUM AMVA ASCAT DES

1.19 1.21 1.21 1.21

0.60 0.61 0.61 0.61

0.60 0.61 0.61 0.61

1.19 1.21 1.21 1.21

ASUM AMVA ASCAT DES

0.64 0.64 0.63 0.64

0.47 0.47 0.47 0.47

0.70 0.68 0.69 0.68

1.19 1.21 1.21 1.21

Node Ai

E

In Table 10.33 we compare the results for the throughputs Xi and the mean number of jobs xi. For this example, the results of all methods are close together and close to the DES results. As in the case of asymmetric single station queueing networks, the preceding approximations are accurate if the values of the service rates in a station do not differ too much. As a rule of thumb, we consider the approximations to be satisfactory if pmin/pmax < 10 at each asymmetric node. Problem 10.6 Formulate the closed network of Fig. 10.57 as a GSPN and solve it exactly using SHARPE or SPNP. Compare the exact results with the approximative onesobtained in this section using ASUM, AMVA and ASCAT.

10.8

NETWORKS

WITH

BLOCKING

Up to now we have generally assumedthat all nodes of a queueing network have queues with infinite buffers. Thus a job that leaves a node will always find an empty place in the queue of the next node. But in the case of finite capacity queues, blocking can occur and jobs can be rejected at a node if the queue is full. Queueing networks that consist of nodes with finite buffer capacity are called blocking networks. Blocking networks are not only important for modeling and performance evaluation of computer systems and computer networks, but also in the field of production line systems. Exact results for general blocking networks can only be obtained by using the generation and

548

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

numerical solution of the underlying CTMC (either directly or via SPNs) that is possible for medium-sized networks. For small networks, closed-form results may be derived. The most common techniques presented in the literature for the analysis of large blocking networks are, therefore, approximate techniques. 10.8.1

Different

Blocking

Types

In [Perr94] three different types of blocking are introduced: 1. Blocking after service (see Fig. 10.58): With this type of blocking, node i is blocked when a job completing service at node i wishes to join node j that is already full. This job then has to stay in the server of node i and block until a job leaves node j. This type of blocking is used to model production line systems or external I/O devices [Akyi87, Akyi88b, Akyi88a, Alti82, AlPe86, OnPe89, PeA186, Perr81, PeSn89, SuDi86]. In the literature, this type of blocking has been referred to by a variety of different terms such as Type-l blocking, transfer blocking, production blocking, and non-immediate blocking. Job blocks

Fig. IO.58

node

i

Blocking

after service.

2. Blocking before service (seeFig. 10.59): The job to be served next at node i determines its source node j before it enters service at node i. If node j is already full, the job cannot enter the server of node i and node i is therefore blocked. Only if another job leaves node j will node i be deblocked and the first job in the queue served [BoKo81, GoNe67b, KoRe76, KoRe78]. In the literature, this type of blocking has also been referred to as Type-2 blocking, service blocking, communication blocking, and immediate blocking. Job blocks

node

Fig. 10.59

i

Blocking

before service.

3. Repetitive blocking (seeFig. 10.60): If a job that has been served at node i wishes to go to node j whose queue is full, it is rejected at node j and goes to the rear of node i queue. This procedure is repeated until a job leaves node j, thereby creating one empty slot in the queue of

NETWORKS

WITH

BLOCKING

549

node j. This type of blocking is used to model communications networks or flexible production line systems [Akvo89, BaIa83, Hova81, Pitt79]. Other terms for this type of blocking are Type-3 blocking or rejection blocking.

Fig.

10.60

Repetitive

blocking.

In [OnPe86], different kinds of blocking are compared with each other. In [YaBu86], a queueing network model is presented that has finite buffer capacity and yet is not a blocking network. Here a job that would block node i because node j is full is buffered in a central buffer until there is an empty spot at node j. Such models are especially interesting for production line systems. 10.8.2

Product-Form

Solution

for Networks

with

Two

Nodes

Many different techniques for the solution of blocking networks are available. We only consider a simple exact solution technique in detail that can be applied to closed blocking networks with only two nodes and blocking after service [Akyi87]. This method can also be extended to queueing networks with several nodes, but then it is an approximate technique that is very costly and it can only be used for throughput analysis [Akyi88b]. We consider queueing networks with one job class, exponentially distributed service times, and FCFS service strategy. Transitions back to the same node are not possible (pii = 0). Each node has a fixed capacity iL& that equals the capacity of the queue plus the number rni of servers. Nodes with an infinity capacity can be taken into consideration by setting A4i > K so that the capacity of the node is bigger than the number of jobs in the network. The overall capacity of the network must, of course, be bigger than the number of jobs in the network for it to be a blocking network: K<Ml+M2.

(10.177)

In the case of equality in the preceding expression, deadlocks can occur [Akyi87]. Th e p ossible number of states in the underlying CTMC of a closed network with two nodes and unlimited storage capacity is: Z=K+l.

(10.178)

At first we consider the simple case where the number of servers at each node is ml = m2 = 1. The state diagram of the CTMC for such a two-node network without blocking is shown in Fig. 10.61.

550

ALGORITHMS

Fig. 10.61

CTMC

FOR NON-PRODUCT-FORM

NETWORKS

for a two-nodenetwork without blocking (ml =

m2

= 1).

As can be seen, the minimum capacity at each node is A4i = K (the queue must contain at least K - 1 spots) so that all states are possible. If the nodes have a finite capacity AL&< K, then not all K + 1 states of Fig. 10.61 are possible. Possible states of the blocking network are then given by the condition that the number of jobs at a node cannot be bigger than the capacity of that node. In the case of a transition to a state where the capacity of a node is exceeded, blocking occurs. The corresponding state is called a blocking state. Recall that due to Type-l blocking, the job does not switch to the other node but stays in the original node. The state diagram of the blocking network is shown in Fig. 10.62. P2

P2

P2

Fig. 10.62 CTMC for the two-nodeblockingnetwork (statesmarkedwith * are blocking states)ml = m2 = 1 (blocking after service).

The state diagram of the blocking network shown in Fig. 10.62 differs from the state diagram shown in Fig. 10.61 in two aspects. First, all states where the capacity is exceeded are no longer necessary and, second, Fig. 10.62 contains the blocking states marked with an asterisk. The number of states 2 in the blocking network is the sum of all possible non-blocking states plus all the blocking states: i = min{K, hli + 1) + min{K, AL?2 + 1) - K + 1.

(10.179)

It can be shown [Akyi87] that for this closed network with two nodes and blocking after service, an equivalent closed network with two nodes without blocking exists. Apart from the number of jobs i? in the equivalent network, all other parameter values remain the same. The population k in the

NETWORKS

equivalent network

can be determined

WITH

BLOCKING

551

using Eqs. (10.178) and (10.179):

I? = min{ K, IHr + 1) + min{K,

(10.180)

A/IQ + 1) - K.

The CTMC of the equivalent network isshown in Fig. 10.63. The equivalent network is a product-form network with Ici jobs at node i, 0 5 & 5 I?.

Fig.

10.63

CTMC

of the equivalent network corresponding

to the CTMC

in Fig. 10.61.

The steady-state probabilities of the equivalent network can be determined using the following equation: (10.181) By equating the steady-state probabilities of the blocking network with the corresponding steady-state probabilities of the equivalent non-blocking network, the performance measuresof the blocking network can be determined. To demonstrate how this technique works, the following simple example is used: Example 10.19 Consider a closed queueing network with two nodes, K = 10 jobs, and the routing probabilities: PI2 = 1,

P21 =

1.

All other system parameters are given in Table 10.34. Table IO.34

1 2

System parameters

1 1

2 0.9

for Example

1 1

10.19

7 5

All service times are exponentially distributed and the service discipline at each node is FCFS. Using Eq. (10.178), the number of states in the network without blocking is determined to be 2 = K + 1 = 11. The CTMC of the network without blocking is shown in Fig. 10.64, while the state diagram of the blocking network (seeFig. 10.65) contains all possible non-blocking states

552

ALGORITHMS

Fig. IO.64

FOR NON-PRODUCT-FORM

CTMC

of the Example

NETWORKS

10.19 network

without

.___-_______

blocking.

-

Fig. 10.65

CTMC

of the Example

10.19 network

-

-

- -

_ _ -

_ _ _ I

with blocking.

from the original network as well as the blocking states (marked with asterisk). The number of states 2 of this blocking network is given by Eq. (10.179) as the sum of all possible non-blocking states (= 3) and the blocking states (= 2): 2 = 5. The CTMC of the equivalent network (see Fig. 10.66) therefore contains 2 = 5 states and the number of jobs in the network is given by Eq. (10.180) as I? = 4.

Fig. 10.66 The throughput ly using the MVA network:

CTMC

of the equivalent

network

for Example

X,,(4) of the equivalent network and is equal to the throughput

A,( 10) = X,,(4) The steady-state probabilities ing with the corresponding blocking network:

10.19.

can be determined directX~(10) of the blocking

= 0.500.

of the blocking network steady-state probabilities

~(6,4)

= n(2,2)

= 0.120,

7r(7,3)*

= n(4,O)

= 0.550,

n(5,5)

7r(7,3)

= 7r(3,1)

= 0.248,

n(5,5)*

= n(l, = ~(0,4)

are obtained by equatof the equivalent non-

3) = 0.054, = 0.026.

NETWORKS

WITH

553

BLOCKING

The mean number of jobs at both nodes can be determined using the steadystate probabilities: K1 = 7 [7r(7,3)* + 7r(7,3)] + 6r(6,4) + 5 [7r(5,5) + 7r(5,5)*] = 6.73, K2 = 3 [7r(7,3)* + n(7,3)] + 4~(6,4) + 5 [7r(5,5) + 7r(5,5)*] = 3.27. With these results all other performance measures can be derived using the well-known formulae. Of special interest for blocking networks are the blocking probabilities: PB1 = n(5,5)* = 0.026, PB2 = 7r(7)3)* = 0.0550. Next consider the case where each node has multiple servers (mi > 1). The state diagram of the CTMC underlying such a two-node network without blocking is shown in Fig. 10.67, while the state diagram of the blocking

Fig. IO.67

CTMC

of a two-node network without blocking (m, > 1).

network (with Type-l blocking) is shown in Fig. 10.68. Here a state marked m2

P2

",2

-/---A+**..*/-*...*

2P2

(m2--1)C12 \

*

7n2P2

m2fi2 /---.+

l

mlP1

mlP1

mlP1

(ml-l)fil

2@1

‘*

Pl

Fig. IO.68 CTMC for the two-node network with blocking. Blocking marked with * (mi > 1).

states are

with asterisks shows the number of blocked servers in that state. Therefore we get for the number fi of states in the blocking network, the sum of all possible non-blocking states plus the blocking states (mi > 1): 9 = min{K, 1Mr + rnz} + min{K, Mz + ml} - K + 1.

(10.182)

I? is given by Eqs. (10.178) and (10.182): I? = min{K, Mr + m2) + min{K, Mz + ml} - K.

(10.183)

554

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

The state diagram of the equivalent network is shown in Fig. 10.69. The equivalent network is again a product-form network with & jobs at node i, 0 5 ii 5 k. P2

ml/4

Fig. IO. 69

m2p2

21-12

2Pl

mlpl

CTMC

of the equivalent

map2

product-form

CL1

network

(m, > 1).

The steady-state probabilities of the equivalent network can be determined using Eq. (7.59): (10.184) with pi(&), Eq. (7.62). The following example is used to demonstrate this technique: Example 10.20 Consider again the network given in Example 10.19, but change the number of servers, m2 = 2. All other network parameters remain the same. From Eq. (10.178), the number of states in the network without blocking is 2 = K + 1 = 11. The corresponding state diagram is shown in Fig. 10.70. The state diagram of the blocking network (see Fig. 10.71)

Fig. 10.70

CTMC

of the Example

10.20 network

without

blocking.

contains all possible non-blocking states from the original network plus all the blocking states (marked with *). The number of states 2 of the blocking network is given by Eq. (10.182) as the sum of all non-blocking states (= 3) and the blocking states (= 3): 2 = 6. The state diagram of the equivalent network (see Fig. 10.72) contains, therefore, also 2 = 6 states, and for the number of jobs in the network, Eq. (10.180) is used: I? = 5. The throughput X,,(5) of the equivalent network can be determined using the MVA and, because of the equivalence of both networks, is identical with the throughput X~(l0) of the blocking network: X,(10) = X,,(5)

= 0.499.

NETWORKS

I I

Fig. IO.71

CTMC

P2

of the Example

2P2

Fig. 10.72

CTMC

10.20 blocking

2P2

of the Example

Pl

1

__-___--'

network.

a-42

10.20 equivalent

555

BLOCKING

Pl

Pl

Pl

Pl

t-------_----__-_______

WITH

product-form

2P.2

network.

The state probabilities of the blocking network are given by equating the corresponding steady-state probabilities of the equivalent non-blocking network. The steady-state probabilities are determined using Eq. (10.181): 7r(7,3)** = 7r(5,0) = 0.633,

n(6,4) = n(2,3) = 0.014,

7r(7,3)* = 7r(4,1) = 0.284, 7r(5,5) = 7r(l, 4) = 0.003, n(7,3) = x(3,2) = 0.064, n(5, S)* = 7r(0,5) = 0.001. The mean number of jobs at each node can be determined using the steadystate probabilities: Kl = 7 [7r(7,3)** + 7r(7,3)* + 7r(7,3)] + 6X(6,4) + 5 [n(5,5) + n(5,5)*] = 6.978, z,

= 3 [7r(7,3)** + n(7,3)* + 7r(7,3)] + 4n(6,4) + 5 [7r(5,5) + n(5,5)*] = 3.022.

All other performance measures can be determined using the well-known equations. Of special interest are blocking probabilities: P& = 7r(5,5)* = 0.001, P&

= 7r(7,3)** + $(7,3)’

= 0.775.

For computing PB~,the steady-state probability ~(7,3)* can only be counted half becausein state (7,3) * only one of the two servers of node 2 is blocked. For blocking networks with more than two nodes, no equivalent non-blocking network can be found. For these casesonly approximate results exist, which

556

ALGORITHMS

FOR NON-PRODUCT-FORM

NETWORKS

are often inaccurate or can only be used for very special types of networks such as tandem networks [SuDi84, SuDi86] or open networks [PeSn89]. Problem 10.7 Consider a closed blocking network with N = 2 nodes and K = 9 jobs. The service times are exponentially distributed and the service strategy at all nodes is FCFS. All other network parameters are given in the Table 10.35. Table 10.35 Network Problem 10.7

i

ea

1 2

1 1

input parameters

for

l/Pi

mi

Mi

1 1

1 1

6 4

(a) Draw the state diagram for the network without blocking as well as the state diagram of the blocking network and the equivalent non-blocking network. (b) Determine the throughput XB, the mean number of jobs xi, blocking probabilities I& (i = 1,2) for the blocking network.

and the

Problem 10.8 Directly solve the CTMC of the blocking network in Problem 10.7 by using SHARPE. Note that you can easily obtain transient probabilities and cumulative transients with SHARPE in addition to the steady-state probabilities. Problem 10.9 For Problem 10.8, construct an SRN and solve it using SPNP. Compare the results obtained from SPNP with those obtained by hand computation. Note that SRNs enable computation of transients and cumulative transients. Furthermore, easy definition of rewards at the net levels enables straightforward computation of all desired performance measures.

Optimization

Analytic performance models are very well suited as kernels in optimization problems. Two major categories of optimization problems are static and dynamic optimization. In the former, performance measures are computed separately from an analytic queueing or CTMC model and treated simply as functions (generally complex and non-linear) of the control (decision) variables. In the latter class of problems, decision variables are integrated with the analytic performance model and hence optimization is intimately connected with performance evaluation. We limit our discussion to static optimization. For a treatment of dynamic optimization in the context of computer and communication systems see [PGTP96], [DrLa77], [MeDii97], [MTD94], [MeFi97], [MeSe97], and [Meer92]. Designs of early computer and communication systems made little use of mathematical optimization techniques. Because the systems were not so complex, the number of possible design alternatives was limited and a good design decision was often obvious. This state of affairs has changed dramatically since modern systems are very complex and requirements for high performance and dependability have evolved. Therefore an optimal design is not obvious any more and the number of possible design decisions can be enormous. A system designer is called on to produce a design of an implementable system from a given design specification. The design specification may include cost, performance, and dependability specifications, in addition to functional specifications. The performance specification states how well the specified functions (as stated in the functional specification) should perform in terms of throughput, response time, etc. The dependability specification states how the system tolerates component/sub-system faults. The dependability spec557

558

OPTIMIZATION

ification also includes minimum acceptable reliability, availability, etc. Cost specifications may indicate maximization or minimization of total cost or provide a constraint in the optimization. The system design methodology takes the system specification as an input and transforms it into a system configuration (characterized by a set of parameters) that fulfills the specification. These (hardware) parameters include the number of processors, the number of memories, the speed and capacity of resources, and other architectural parameters such as the system topology. Software configuration parameters describe the distribution of processes over system resources as well as resource scheduling policies. In addition to the hardware and software specification, maintenance policies might be described (e.g., preventive vs. corrective). It is thus a very difficult task to choose the configuration that best meets the system specification. Traditionally, dependability, performance, and economic analyses of computer systems have been studied separately and independently. Optimization methods provide a feasible avenue for combining performance, dependability, and cost for deriving optimal system parameters that meet the stated design specifications. Usually these questions are multiobjective optimization problems. Although it is possible to deal directly with multiobjective problems [IPM82], it is common to pick a single relevant system specification to be used as the objective function and to let the remaining design factors provide constraints. An alternative is to combine multiple criteria into a single utility function.

11.1

OPTIMIZATION

PROBLEMS

AND COST FUNCTIONS

To formulate the optimization problem it is important to determine the system parameters that can be varied and hence considered to be decision variables. Possible types of optimization are: l

Minimization

of cost for given throughput.

l

Maximization

of throughput

l

Minimization

of the mean response time with cost constraint.

with cost constraint.

The simplest cost functions are linear functions: If device (resource) service rates /.~i are decision variables, the linear cost function is given by: C(p) = 2

tip; = COST,

(11.1)

i=l

with the vector of the service rates p = (~1,. . . , pi). COST is the total cost of the considered system, and ci is the cost factor for service rate pi of resource i.

OPTIMIZATION

BASED ON THE SUMMATION METHOD

559

More realistic estimates are obtained by using non-linear cost functions: (11.2) The service rate CL;of a server depends on the speed vi of device i and the mean number of work units di per visit:

For the CPU, the mean number of instructions per CPU burst is given by di and the CPU speed (number of instructions per time unit) by vi. In the case of a storage device, di is the mean number of words in an I/O operation and 21;is the speed of the storage device in words per time unit [TrKi80]. Thus if the mean number of work units di of the jobs at station i is given, then the optimal value of the device speed vi is immediately given using the optimal values of the service rates pi. We can extend the optimization problem if we also consider the cost for main memory. In the case of a multiprogramming computer system, this cost depends on the degree of multiprogramming K (the number of jobs in the system). The cost function is then given by: C(/J, K) = C(K) + 5

(11.3)

cipyz = COST

i=l

where the C(K) is the cost of the main memory satisfying the storage requirement of K concurrent jobs. The next step is to choose the performance model. This can be a DES model or an analytic model. Among the latter we can choose a product-form queueing network model or a non-product-form queueing network model, a CTMC model, or a hierarchical model. The final step is to choose the optimization method. There exist numerous optimization methods that can be used in combination with analytical methods for the performance evaluation introduced in this book. We show the use of a solution method based on Lagrange multipliers. We introduce an approximation method based on SUM (see Section 9.2) and a more complex but exact method, which is based on the convolution method (see Section 8.1). Finally, we show the use of geometric programming in the design of a memory hierarchy. For further study of performance optimization including the issue of convexity, see [TrKi80], and [SRT88].

11.2

OPTIMIZATION

BASED ON THE SUMMATION

METHOD

This method was introduced by Belch, Fleischmann, and Schreppel [BFS87] (therefore called BFS method for short) and is now demonstrated using the following example:

560 11.2.1

OPTIMIZATION

Maximization

of the Throughput

We assume a closed queueing network with N stations and K jobs of a single class. The visit ratios e; and the functions fi (see Eq. (9.15)) that describe the behavior of the queues are also given. The correction factor (K - 1)/K is neglected to obtain formulae that are as simple as possible. This approach is valid if the number of jobs is large, because then (K - 1)/K is close to 1. However, usable formulae are also obtained if the correction factor is included. These provide more accurate values for the optimum, especially for small values of K [AkBo88b]. For Xi = X . ei, rni = 1 and pi = $ we obtain: Type-l, 2,4, and mi = 1, Ki(X9P.i)

=

fi(X,/%)

=

Type-3 IS. We seek service rates pi that maximize the overall throughput X and satisfy a linear cost constraint: C(p) = g,,i

= COST +

i=l

Here the cost C(K) for the main memory is neglected but can easily be included by repeating the optimization for several values of K and choosing the one with the maximum throughput . The service rates must satisfy the system equation as well as the cost constraints. The system equation is as follows, using Eqs. (9.16) and (9.22):

The optimization problem can be solved using Lagrange multipliers [BrSeSl]. The Lagrange function L(X, ~1,. . . , PN, yi, 92) with objective function x and two Lagrange multipliers yi and y2 is given by: N

q~,Pl,...,

~N,yl,!/2)=~+~1

CG/X-COST i=l Xei +Y2

c(

#IS

+$-” Pi

-

. i

Xei

)

A necessary condition for optimal service rates pi and maximum throughput X is obtained by differentiating with respect to X7 Pi7 Yl, Y2: dL ax=

E=o @i

-= dL 0

0, ’

8Yl

i=l

aL ,“‘7 N ? -= dY2

(11.4)

’

(11.5)

0 *

OPTIMIZATION

BASED ON THE SUMMATION METHOD

561

The following optimal values are obtained by solving the preceding system of equations:’ Jj* =

COST *K

(11.6)

(~~)‘+K~w

type i # IS, (11.7) type i = IS. It can be seen from Eq. (11.6) that the maximum throughput A* is directly proportional to the total cost COST. The optimal service rates pf cannot be explicitly stated in more complicated optimization problems with non-linear cost functions, several types of stations, and further auxiliary constraints. They can be obtained iteratively from the system of equations. Further results for some of the optimization problems already mentioned are given in the following. 11.2.2

Minimization

of Cost

Next we consider the minimization of cost subject to a minimum throughput requirement, A. By the same method as in Section 11.2.1, we obtain a closedform formula for the approximate optimal values of service rates:

The minimal cost is given by:

C”(p)= &lCi. i=l

‘Since the summation method is an approximation values are approximate optimal values.

method for performance

analysis, these

562

OPTIMIZATION

The following iteration method can be used for a non-linear cost function: pi = 1, for i = 1, . . . , N. Initialize: Iterate until the deviations are sufficiently small: y=x.

($&q2,

(11.9)

i=l

, Jwi + 7a&pqt-1

type i # IS, (11.10)

1 hei ( >

a,--1

,

11.2.3

Minimization

type i = IS.

QiC,

of the Response

Time

The problem of minimizing the mean response time T is equivalent to the problem of maximizing the overall throughput. This result follows from Little’s theorem, T = K/X, and the fact that the number of jobs K in a closed network is constant. Explicit formulae can be provided if the cost function is linear. In the following we wish to maximize the throughput with Example 11.1 cost constraints. The service rates that produce maximum throughput are to be calculated. The example consists of a product-form queueing network with N = 5 stations, three of which are of type -/M/l and two of type -/G/oo. The queueing network contains 20 jobs. The visit ratios are as follows: el = 1, e2 = 0.2, e3 = e4 = 0.5, e5 = 0.3. We assume a linear cost function with the following coefficients: Cl = 10, c;! =

c3

= 5,

c4

= 2,

c5

= 1.

The total cost is constrained to be COST = 100. Using the BFS method, the following values are obtained for approximate optimal service rates that achieve the throughput X* subject to the given cost constraints: /A; = 6.189, /L; = 1.689, /L; = 3.812, & = 1.096, /~uj= 1.236, with the approximate optimal throughput: A* = 6.189. The exact value of the throughput X = 6.569 for the approximate optimal service rates pL;*is obtained by means of MVA. In order to verify these approximate optimal-values, we vary the service rates, subject to the constraint: N c i=l

Ci/..Li

= 100,

OPTIMIZATION

563

BASED ON THE SUMMATION METHOD

as before. The new throughput is then compared with the approximate optimal throughput. If the following values of the service rates are used: p1 = 7, /.L2= 1.7, pg = 3.8, pLLq = 1, pug= 0.5, the value for the throughput obtained by MVA is X = 6.473. This is less than X = 6.569, the value of the throughput obtained using the approximate optimal optimal service rates pa. Similar results are obtained for other values of the service rates that satisfy the constraint given previously. If we increase the distance between the selected service rates and those that produce the approximate optimal throughput considerably: PI

= 8,

then the throughput

P2 = 2,

P3 =

p4 =

2,

CL5 =

1,

decreases and approaches its minimal value X = 2.

Table 11.1 Maximization of the throughput Example BFS Method

1,

1

with cost constraints (COST = 100)

Example 2

Complex Method

BFS Method

Example 3

Complex Method

BFS Method

Complex Method

Pi & & 4 CL:

6.90 1.69 3.81 1.13 1.24

6.81 1.65 3.99 1.01 1.50

8.07 1.30 2.83 0.90 1.10

7.97 1.41 2.95 0.93 1.13

3.04 0.66 1.27 0.59 0.67

3.00 0.69 0.80 0.79 0.84

%FS hNA

6.19 6.57

6.56 6.55

6.66 7.54

7.56 7.57

2.63 3.00

2.95 2.97

100.00

99.81

100.00

99.97

100.00

100.00

COST

Table 11.1 contains additional examples of the maximization of the throughput of a network with linear cost constraints. It compares values obtained by the BFS method with values obtained by the complex method which requires much more computation. If we consider the simplicity of our method, it is surprising how close our values are to those obtained by the complex method. These examples show that optimization problems can be relatively easily solved using the system of equations. If the cost constraint is linear, then the optimal service rates ,LL;can be explicitly stated. If the cost function is non-linear, then a simple iteration method converges well. Table 11.2 contains the results for the optimization of the throughput with a non-linear cost function. The service rates that produce optimal throughput were used as input values for the exact MVA. The BFS method is well suited for solving optimization problems because it provides a simple relation between the quantity to be optimized (e.g., throughput) and the decision variables (service rates). If the cost constraint is linear, then explicit formulae for optimal service rates and throughputs are provided. It is also possible to apply the BFS

564

OPTIMIZATION

Tab/e 11.2 Optimization of throughput the non-linear cost function)

(CX= cxpouent of

a

1

1.25

1.5

2.0

2.5

3

6 CL; Pu5 4 d

6.912 1.689 3.182 1.096 1.236

4.911 1.242 2.727 0.921 0.998

3.884 1.017 2.173 0.821 0.883

2.860 0.806 1.625 0.735 0.781

2.361 0.716 1.363 0.706 0.774

2.067 0.674 1.214 0.704 0.732

%FS xMVA

6.189 6.596

4.438 4.711

3.532 3.752

2.625 2.790

2.180 2.319

1.918 2.039

method to non-product-form queueing networks with arbitrarily distributed service times if we use the corresponding formulae in the system equation (see Section 10.1.4.3).

11.3

OPTIMIZATION ALGORITHM

BASED ON THE CONVOLUTION

With the BFS method, combined with Lagrange multiplier, we obtained approximate results for the optimization problem. To obtain exact results, we can again use the concept of Lagrange multipliers together with the convolution algorithm [TrKi80, SRT88]. 11.3.1

Maximization

of the Throughput

In the convolution algorithm we use the following formula (see Eq. (8.14)) for the throughput:

G7 K) =

G(P7K - 1) G(CL’K) ’

(11.11)

with:

5 ki=K ‘=I

2=1

and:

Pi(h) =

ki!7

ki I mi7

mi!mFierni,

ki 2 mi,

17

mi = 1.

(11.12)

OPTIMIZATION

BASED ON THE CONVOLUTION ALGORITHM

565

As a cost function, we use the one suggested in Eq. (11.2) with additional main memory cost C(K) : N

C(p, IT) = C(K) + c

(11.13)

c~,cL;~= COST.

i=l

To maximize the throughput with a cost constraint, we can use the following Lagrange function, which is derived from Eqs. (11.11) and (11.13)):

-qcL,Y, K)

= X(/A, K) + y

C(K) - COST + 5 tips”

-

i=l

In order to obtain the optimal values for ~1, we differentiate the Lagrange function. This operation results in the following non-linear system of equat ions: 8L G =O, aL ay=

0

i=

l,...,N,

(11.14) (11.15)

*

We can use the computer algebra program MAPLE to obtain the derivatives and solve the non-linear system of equations. Alternatively, Moore’s formula [Moor721 for the normalization constant can be used to obtain explicit formula for the preceding derivatives [TrKi80]. The system of non-linear equations can then be solved using the Newton-Raphson or other suitable method. The solution of the system is the optimum service rate vector ~1that fulfills Eq. (11.15). Th e maximum throughput can then be obtained with Eq. (11.11) Example 11.2 Consider a multiprogramming computer system with a CPU and three disk devices. The product-form queueing network model is given in Fig. 11.1. The corresponding routing probabilities are: Pll

= 0.05,

P12 =

0.5,

p13 = 0.3,

$I14= 0.15,

p21 = p31 = p41 = l,(),

For special values of the cost coefficients ci and the exponents oi (given in Table 11.3), the total budget COST = 500 and a linear memory cost function C(K) = C,K with C, = 50, the optimal values of the throughput X”, the service rates &’ and degree of multiprogramming K are given in Table 11.4. We see that we have an optimum of throughput when degree of multiprogramming K = 2. The computation time for this method is high compared to the BFS method. On the other hand this method delivers exact results. Problem

11.1

Use the BFS method for Example 11.2.

566

OPTIMIZATION

Disk

CPU

Disk

Fig. 11.1

Central-server model of the multiprogramming

system in Example 11.2.

Table 11.3 Cost factors cz and exponents a, for the central-server model of Fig. 11.1

CPU Disk1 Disk2 Disk2

11.3.2

Optimal

Design

131.9 11.5 54.2 54.2

of Storage

0.55 1.00 0.67 0.67

Hierarchies

This example is intended to address the problem of optimally determining the technology and capacity of the different memory levels of a linear storage hierarchy. The storage hierarchy is used in a multiprogrammed environment. Our goal is to maximize the system throughput subject to a cost constraint. It is assumed that we know the number of storage levels as well as the capacity of the slowest level. Consider the storage hierarchy, given in Fig. 11.2 [TrSiSl]. It consists of m electronically accessed levels (numbered 1,2, . . . , m), as for example a cache and p mechanically accessed levels (numbered m + 1, m +

Table 11.4 Optimal throughput A* with the optimal service rates ~5 for the central-server model of Fig. 11.1

1 2 3 4 5

0.071 0.091 0.089 0.076 0.059

3.090 2.819 2.356 1.967 1.557

4.240 3.373 2.684 1.979 1.084

2.054 1.525 1.238 0.847 0.598

1.376 1.032 0.752 0.571 0.431

OPTIMIZATION

BASED ON THE CONVOLUTION ALGORITHM

567

Cost per byte

Access time and capacity

Delay boundary

Fig. 11.2

A

linear storage hierarchy.

2 - - 7m + p), e.g., disk. Let M = m + p. These two groups are divided bk a delay boundary. It is assumed that the speed increases with higher levels (lower numbers denote a faster level) and the capacity decreases with higher levels. Whenever a piece of information is not found in level 1, a request is sent to a lower level. This activity goes on until the request can be satisfied. If i < m (meaning the device is above the delay boundary), it is assumed that the CPU is kept waiting until the information is retrieved, meaning that there will be no more than one request at levels 1,2, . . . , m. For m + 1 5 i 5 m + p, a process swap is performed to allow another process to use the CPU, meaning that several requests may be queued at each of the memory levels m+l, m+2,. . . , m+p. This linear storage hierarchy is modeled using a closed queueing network where K (the multiprogramming degree) programs circulate sharing the resources of the system. The capacity of the ith level is bi bits, which are organized into blocks of si bits each. The block sizes si, i = 1,2,... , M, are assumed to be known. The K active programs share the capacity of each level equally. A success function Hi denotes the probability that a storage reference is a hit at level i. A reasonable assumption is that this function depends only on hi/K and si [TrSi81]. Hi = H&/K,

si)

i = 1,2,. . . ,M.

Let the miss ratio function Fi be defined as & = 1 - Hi. The hit ratio is the probability that a storage reference generated by the CPU is found in

568

OPTIMIZATION

level i and not in levels 1,2, . . . , i - 1. The hit ratio hi is given by:

The definition of bu and so is such that Fe(be/K, SO) = 1 and F~(blll/K, 0. Let pa be the probability that a given request is the last memory i.e., the program has finished execution). Then the average total ( of memory references per program is given by l/pe. The probability memory request at level i is satisfied is then given by: pi=(l-pa)hi,

i=1,2

,...,

SM) = request number that a

M.

The closed queueing network model of this system is shown in Fig. 11.3 [TrSiSl]. All 1evels where requests are not queued are merged into an equiva-

fig. 11.3

Queueing network model of the linear storage hierarchy shown in Fig. 11.2.

lent server. The branching

probabilities

PO0 =

=

are given as:

“; CPi’ i=l

Ipoj

for the network

l?iz

)

j = 1,2 ,...,

p.

1-EPi i=l In [TrSiSl]

the service time for the equivalent

server is derived:

(11.16)

OPTIMIZATION

BASED ON THE CONVOLUTION ALGORITHM

569

In this context tj denotes the average transfer time for a block of size sj-1 from the jth level to the (j - l)st level. Furthermore we assume that the service discipline at all nodes is FCFS and all service times are exponentially distributed. The goal is to maximize the system throughput, which is defined as the average rate of flow along the new program path, X = poo,x~po, where pe is the utilization of the equivalent server node and is given by:

po= s m-4 K - 1)) PO w4K) I-1=(Po,P1.1,...,ccp), x0=;

=gpi2ti+

5

i=l

=

(’

j=l

-PO)

tl

+

pi&j,

i=m+1

j=l

~tj~j-l(bj-l/K,

i

(11.18)

,

Sj-1)

j=2

1

P

1 -

(11.17)

= eiti+,

= ti+m

c

Pi

Pj+-rn,

j=l

=

(1 - PO)ti+mFi+m-l(Ki+m-l/K,

Si+m-l),

i =

1,2,.

. . ,P.

(11.19)

In these expressions, e = (ee, er, . . . , eP) is the vector of the visit ratios of the specified closed queueing network. Since the ei can only be determined within a multiplicative constant, we choose: m

ee-l-

c

Pi,

i=l

which yields: P ei =

c

Pj+m*

j=l

The system throughput

The optimization words, to minimize:

is given by:

problem is to maximize the throughput,

or, in other

Let the technology cost curve for the ith level be given by Ci(ti). The total cost of memory level i is given by ci(ti)Bi (bi), where Oi(bi) accounts for the

570

OPTIMIZATION

economies of scale that are present in the memory t ethnology. The optimization problem can now be defined as follows: Minimize x(p, K) = -1 G(CL,K) POG(CL,K - 1) ’ with p as defined in Eqs. (11.17), (11.18), and (11.19), and subject to: 5

ci(ti)Oi(bi)

5 COST.

(11.20)

i=l

In this case the decision variables are the capacities and the average access timesforeachlevel(bi,t;,i= 1,2 ,..., M). Thevaluesm,p,K,pa,bM,si ,..., SM-1, and COST are assumed to be fixed parameters. Furthermore, we assume that & and 8i are functions of the memory capacities bi. The b;, ci , and 8i are posynomial functions of the access times ti (posynomials are generalizations of polynomials where the coefficients are positive real numbers while the exponents are allowed to be arbitrary real numbers). In the special since: case of K = 1, x(p, 1) is a polynomial function of the ,UU;

The optimization problem in the case of K = 1 is then seen to be a standard geometric programming problem , provided we assume that bi, ci, and 1!3iare polynomial functions of ti. In [TrSi81] it is shown that even in the multiprogramming case, the optimization problem becomes a convex programming problem when logarithmically transforming the variables ti and bi (i.e., ti = e”z and bi = eWz,i = 1,2,. . . , A4) and requiring the functions Fi (bi ) , ci (t; ) and hi(bi), i = 1,2,. . . , M to be posynomial functions. Thus any local solution to the optimization problem is also its global solution.

12 Performance

Analysis Tools

Performance analysis tools have acquired increased importance due to increased complexity of modern systems. It is often the case that system measurements are not available or are very difficult to get. In such cases the development and the solution of a system model is an effective method of performance assessment. Software tools that support performance modeling studies provide one or more of the following solution methods: l

Discrete-event simulation.

l

Generation and (steady-state and/or transient) solution of CTMC and DTMC.

l

Exact and/or approximate solution of product-form

l

Approximate

l

Hierarchical (multilevel) models combining one or more of the preceding methods.

solution of non-product-form

queueing networks.

queueing networks.

If we use DES, then the system behavior can be described very accurately, but computation time and resource needs are usually extremely high. In this book queueing network solutions as well as Markov chain analysis methods have been introduced to analyze system models. Queueing networks are very easy to understand and allow a very compact system description. For a limited class of queueing networks (see Chapters 8 and 9), so-called product-form queueing networks, efficient solution algorithms (such as convolution, MVA, SCAT) are available. But many queueing networks do not fulfill the productform requirements. In this case approximation methods can be used (see 571

PERFORMANCE ANALYSIS TOOLS

572

Chapter 10). It is also possible to develop a multilevel model to approximately solve a non-product-form queueing network. If the approximation methods are not accurate enough or cannot be applied to a certain problem, as can, for example, be the case when we have non-exponentially distributed service times or when blocking is allowed in the network, then DES or CTMC is used. As we have seen in earlier chapters, the state space for even simple systems can be huge and grows exponentially with the number of nodes and the number of jobs in the network. Nevertheless, due to increasing computational power and better solution algorithms, CTMCs have acquired greater importance. For the solution of our models we realize that: l

The application of exact or approximate solution algorithms for queueing networks is very cumbersome, error prone, and time consuming and hence not feasible to carry out by hand calculations.

l

The generation and solution of the CTMC/DTMC tems by hand is nearly impossible.

for even small sys-

We therefore need tools that can automatically generate the state space of the underlying CTMC and apply exact or approximate solution algorithms and/or that have implemented exact and approximate algorithms for queueing networks. In this chapter we introduce four representative performance modeling and analysis tools: PEPSY, SPNP, MOSES, and SHARPE. As mentioned previously, tools differ not only in their solution algorithms (DES, exact and approximate solution algorithms for queueing networks, generation and transient and steady-state solution of CTMCs) but also in their input/output facilities. There are tools that have a graphical user interface (GUI) and/or a special input language (batch mode or interactive) or both. Some tools are based on special model types such as queueing networks, SPNs, CTMC models, or precedence graphs (and others) or a combination of these model types.

12.1

PEPSY

PEPSY (performance evaluation and prediction system) [BoKi92] has been developed at the University of Erlangen-Niirnberg. Using this tool it is possible to describe and solve PFQNs and NPFQNs. More than 30 solution algorithms are incorporated. It is possible to specify open, closed, and mixed networks where jobs cannot switch to another job class. Closed job classes are described by the number of jobs in each class, while open job classes are described by the arrival rate of jobs at the class. To compute performance measures such as throughput, utilization, or mean response time of a given network, different exact as well as approximation algorithms are provided for PFQN and NPFQN. The network is described in textual and/or graphical form. The X11-windows version of PEPSY is called XPEPSY [Kirs93]. PEPSY

PEPSY

573

can be used on almost all UNIX machines, whereas XPEPSY needs the X11 windows system. A Windows95, WindowsNT version WinPEPSY with similar features is also available but it has a restricted set of solution algorithms. 12.1.1

Structure

of PEPSY

12.1.1.1 The input File The PEPSY input file contains the specification of the system to be modeled. The specification file has a name of the form e-name and is divided into a standard block and possibly additional blocks that contain additional information specific to solution algorithms. The standard block contains all necessary descriptions for most of the solution methods, for example, the number of nodes and job classes, the service time distribution at each node, the routing behavior of the jobs, and a description of the classes. The input file is usually generated using the input program, which is provided by PEPSY. Information that is not asked for by the input program is provided using the addition program. The type of each node (and its characteristics) needs to be specified: the -/M/m-FCFS, -/G/l-PS, -/G/-IS, and -/G/l-LCFS) and several other node types. For example, two types of multiple server nodes with different service time distributions -/M/m-FCFS-ASYM and -/G/m-FCFS-ASYM can be specified. The service time distribution is defined by its first and second moment, or by the service rate and the squared coefficient of variation. For the solution of the specified models, many different algorithms are implemented in PEPSY. These algorithms can be divided into six groups:

1. Convolution algorithm for product-form 2. MVA for product-form

networks.

networks.

3. Approximation

algorithms for product-form

4. Approximation

methods for non-product-form

networks. networks.

5. Automated generation and steady-state solution of the underlying CTMC. 6. DES.

In addition to these six groups, there are some methods and techniques that do not fit in any of these groups. For example, the bounds method performs a calculation of the upper bounds of the throughput and lower bounds of the average response time. 12.1.1.2 The Output file For each file name e-name, an output file with the computed performance measures is generated. It is called xx-name where “name” is the same as in the input file and “xx” is an abbreviation for the used method. The output file consists of a short header with the name of

574

PERFORMANCE ANALYSIS TOOLS

the model, the corresponding input file, and the solution method used to calculate the performance measures. The performance measures of all nodes are separated by classes. The output file contains the performance measures for each node (throughput, visit ratio, utilization, average response time, average number of jobs, average waiting times, and average queue length) and the performance measures for the entire network (throughput, average response time and average number of jobs). 12.1.1.3 Control Files As already mentioned, PEPSY offers more than 30 solution methods to compute performance measures of queueing networks. In general, each method has certain limitations and restrictions. Because it would be impossible for the user to know all these restrictions and limitations, control files are used in PEPSY. Each control file contains the limitations and restrictions for each solution method, namely the network type (open, closed, or mixed), the maximum number of nodes and classes allowed, whether the method can deal with routing probabilities or visit ratios, whether the service rates at -/M/m-FCFS nodes have to be the same for different job classes, the range of the coefficients of variation, and the maximum number of servers at a -/M/m node. 12.1.2

Different

Programs

in PEPSY

The following lists the most important detailed reference see [Kirs94]. input:

programs of PEPSY. For a more

This program is used for an interactive description of a queueing network.

addition: For some solution algorithms, additional information is needed. This information is provided by invoking the addition program. selection: Using this program, all possible solution algorithms can be listed that are applicable to the system description (the file that was just created). The output of this program consists of two columns. The left-hand column contains all solution algorithms that can be applied directly, while the right-hand column contains the methods that can only be applied after specifying an additional block (using the addition program). analysis: After a complete system description, the queueing network desired solution algorithm can be applied by invoking the analysis program. The results are printed on the screen as well as written to a file. pinf:

This program is invoked to obtain information about the solution method utilized.

PEPSY

575

transform: As already mentioned before, it is possible to specify the network using either routing probabilities or visit ratios. By invoking the transform program, visit ratios can be transformed into routing probabilities. pdiffz By invoking pdiff, the results of two different solution methods can be compared.

Fig. 12.1

12.1.3

Example

Simple queueing network example.

of using PEPSY

The main steps in working with PEPSY are described herein. Starting with the model, first the model file is created, then the solution method is selected, and the analysis is carried out. The model to be analyzed is shown in Fig. 12.1. At first the input program is invoked by means the command input. The user will then be asked to enter the type and number of job classes (a closed network with only one job class in our case), the number of nodes (three in our case), the type of each node, and its service rate. After the basic network information is specified, the routing information is entered. In PEPSY we need to consider one special point when talking about routing: open job classes contain a (combined) source/sink-node that is used in PEPSY as a reference node (outside node) for relative performance values. For closed queueing networks the outside node also needs to be specified. Finally the number of jobs in the network needs to be specified. The corresponding model Now the file is saved in, say, e-first-example. description, generated by input, is shown in Fig. 12.2. The next step is the selection of the solution method. PEPSY supports the user with a built-in database with the limitations and restrictions of each algorithm and returns a list of applicable solution methods. In our example the result, of applying the selection program to the model description is shown in Fig. 12.3. This list contains all the methods that can be used to analyze the network e-simple-example. In addition to the standard network description that has been entered in the beginning, some methods need further parameters.

576

PERFORMANCE ANALYSIS TOOLS

fig. 12.2

Fig. 12.3

Model description

The list of applicable

solution

file e-simple-example.

methods

presented

by the selection

program.

577

PEPSY P

\ PERF?ORMANCE,INDICES FORNET: simple,exaqKle description of the network is In file ~e,slmpSe,example' the closed network was 801ved using z;he method 'mari@' jobclass 1 maa l/mu rho I lambda B llWZ marle -----------*---------------------------------------------------------------0.613 0.166 3.922 I.000 0.100 0,392 CPU 3.503 0.700 0,333 0,915 1.276 disk I 2.746 5.883 s.000 o.ooa 6 .a00 1.177 Q*300 tarminaX I characteristic indices: t lambda marie -----------$--------------*-----------mvz I

3.922

2.549

mwsl 0.221 2.588 0.000

maa 10.000

legend 6 : average number of visits rho : utilisation lllV2 : average respbnae time maa : average number of jab8 mwz : averaGe waiting time snws2: average queue-length

Fig. 12.4

mwz 0.056 0.943 0.000

mu : smvice rate lambda: mean throughput

/

Performance measuresof the network esimple-example.

For example, in order to run the sim2 method (DES), some information about the maximum simulation time and accuracy needs to be specified. If the user is not familiar with a particular method, PEPSY assists the user with online help. Suppose the user selects the method name marie and invokes the solution method by entering the command analysis marie simple-example. The resulting performance measures are shown on the screen and are also written into the file a-c-simple-example. The contents of this file are shown in Fig. 12.4. 12.1.4

Graphical

User Interface

XPEPSY

To demonstrate how to work with XPEPSY, the same queueing model as in Section 12.1.2 is used. After the program is started, the user can graphically draw the network as shown in Fig. 12.5. Nodes or connections can be created, moved, or deleted simply by mouse clicks. The necessary parameters are specified in the corresponding dialog boxes. Consider, for example, the node data dialog box, shown in Fig. 12.6~ that appears when clicking on the service station in the drawing area and that is used to specify node parameters. When the queueing network is fully specified, XPEPSY switches from the edit mode to the analysis mode and a new set of menus will appear. Now PEPSY assists you in selecting the solution method. Four possibilities are offered: 1. Select from a limited set of methods, i.e., the ones that are very well suited for the problem. 2. Select from all possible methods.

578

PERFORMANCE ANALYSIS TOOLS

1 22

~~~Wgr

Project w

Mode

Optlons

About

Help1

-----

1’ _ -__“_x”“-_‘L ._

Crld:

50 -J ---I

-.--.”

..--- -“-“.--_--

_--. “““-xI

Working field

_-.--A--

_--_ -‘-.--.~-..-“-2~.-~-.---_~-x_L.-.-.-’-

Standard J ] / Class 1 -A

J

/ Dtrectlon

9

/ Model-name

first

fig. 12.5 The main XPEPSY window.

--

Nods Data

ode name IEPLI --~~---“--~-

,~ ,,I. * I.I, ~::’ 2 2’

Node type Number of nodes

XPEPSY

(b) /

--

I

Class 1

TransItIon Probablhtles

-I 1

2 DISK b 7 r-----~ 3 TERM $1.3 r-

-*1 I----

iI

I”““” Analysis “x” ” , “” * Standard v’ Extended v‘ Results v Non feasible

L-

“-

”

x”x

” ~x~“.~-~xI”“I”“”

Selected method [yilzr--------

M/G/7 - PS

~-“-“--

Method lnformatlon

-m-v---

Analysis

method options 2: :: a’ I : j: F-------numerical parameters

Reget Value_d --

APP’Y

j I

r-

Fig. 12.6 (a) The node data dialog box and (b) the menu of solution methods corresponding to Fig. 12.5.

SPNP

579

3. Show only methods that have already been used and for which valid results can be found without resolving the model. 4. Show all methods that cannot be used due to inherent restrictions of the solution method. Now the solution can be carried out by simply activating the Analysis button from the analysis menu, shown in Fig. 12.6b, and the results are presented on the screen (see Fig. 12.4). The results can also be presented as a histogram or as a line drawing. In a line drawing it is also possible to show an output parameter as a function of an input parameter. Some additional programs from PEPSY are integrated into XPEPSY as well. Using the Method Information button, the PEPSY internal database can be searched for information about the selected method.

12.2

SPNP

The stochastic Petri net package (SPNP) d eveloped by Ciardo, Muppala and Trivedi, is a versatile modeling tool for performance, dependability, and performability analysis of complex systems. Input models developed based on the theory of stochastic Petri nets are solved by efficient and numerically stable algorithms. Steady-state, transient, cumulative transient, time-averaged, and up-to-absorption measures can be computed. Parametric sensitivity analysis of these measures is efficiently carried out. Some degree of logical analysis capabilities is also available in the form of assertion checking and the number and types of markings in the reachability graph. Advanced constructs available - such as marking-dependent arc multiplicities, guards, arrays of places and transitions, and subnets - reduce modeling complexity and enhance the power of expressiveness of the package. The most powerful feature is the capability to assign reward rates at the net level and subsequently compute the desired measures of the system being modeled. The model description language is CSPL, a C-like language, although no previous knowledge of the C language is necessary to use SP N P. 12.2.1

SPNP Features

The input language of SPNP is CSPL (C-based stochastic Petri net language). A CSPL file is compiled using the C compiler and then linked with the precompiled files that are provided with SPNP. The full power of the C programming language can be used to increase the flexibility of the net description. An important feature of CSPL is that it is a superset of C. Thus a CSPL user can exploit C language constructs to represent a large class of SPNs within a single CSPL file. Most applications will only require a limited knowledge of the C syntax, as predefined functions are available to define SPNP objects.

580

PERFORMANCE ANALYSIS TOOLS

Another important characteristic of SPNP is the provision of a function to input values at run time, before reading the specification of the SPN. These input values can be used in SPN P to modify the values of scalar parameters or the structure of the SPN itself. Arrays of places and transitions and subnets are two features of SPNP that are extremely useful in specifying large structured nets. A single CSPL file is sufficient to describe any legal SPN, since the user of SPNP can input at run time the number of places and transitions, the arcs among them, and any other required parameters. The SPNP package allows the user to perform steady-state, transient, cumulative transient, and sensitivity analysis of SPNs. Steady-state analysis is often adequate to study the performance of a system, but time-dependent behavior (transient analysis) is sornet imes of greater interest: instantaneous availability, interval availability, and reliability (for a fault-tolerant system); response time distribution of a program (for performance evaluation of software); computation availability (for a degradable system) are some examples. Sensitivity analysis is useful to estimate how the output measures are affected by variations in the value of input parameters, allowing the detection of system performance bottlenecks or aiding in design optimization. Sophisticated steady-state and transient solvers are available in SPNP. In addition, the user is not limited to a predefined set of measures: Detailed expressions reflecting exactly the measures sought can be easily specified. The measures are defined in terms of reward rates associated with the markings of the SPN (see Section 2.2.3). The numerical solution methods provided in the package address the stiflness problems often encountered in reliability and performance models (see Section 5.1.4.2). A number of important Petri net constructs such as marking dependency, variable cardinality arc, guards, arrays of places and transitions, subnets, and assertions facilitate the construction and debugging of models for complex systems. A detailed description of SPNP and the input language CSPL can be found in [CFMT94] and [CTM89]. SPNP has been installed at more than 100 sites and has been used to solve many practical problems at Digital Equipment Corporation, Hewlett Packard, Nortel and other corporations. Some very powerful new features that have been added to S PN P include the capability of DES of non-Markovian nets and fluid stochastic Petri nets [CNT97]. 12.2.2

The CSPL

Language

Modeling with SPNP implies that an input file describing the system structure and behavior must be prepared. Such an input file can be prepared automatically via the graphical user interface that has been recently developed as discussed in Section 12.2.3. Alternatively, the user may decide to prepare such a file himself. The language designed to do so is named CSPL, a superset of the C language [KeRi78]. What distinguishes CSPL from C is a set of predefined functions specially developed for the description of SPN

SPNP

Fig. 12.7

581

Basic structure of a CSPL input file

entities. Any legal C construct can be used anywhere in the CSPL file. All the f scanf, log, exp, etc., are available and C library functions, such as fprintf, perform as expected. The only restriction to this generic rule is that the file should not have a main function. In spite of being a programming language, CSPL enables the user to describe SPN models very easily. There is no need to be a programmer to fully exploit all the built-in features of SPNP. Just a basic knowledge of C is sufficient to describe SPNs effectively; although, for experienced programmers, CSPL brings the full power and generality of the C language. Fig. 12.7 shows the basic structure of a CSPL input file with six different segments. Parameters Segment The function parameters allows the user to customize the package. Several parameters establishing a specific behavior can be selected. The function iopt (fopt) enables the user to set option to have the integer (double-precision floating point) value. Any of the available options can be selected and modified. For example:

specifies that the solution method to be used is transient solution as uniformization and that the reachability graph is to be printed. The function input permits the input of parameter values at run time. A/et Segment The function net allows the user to define the structure and parameters of an SPN model. For built-in functions that can be used inside the net segment, Fig. 12.8 provides some illustrations.

582

PERFORMANCE ANALYSIS TOOLS

Fig. 12.8

The CSPL

net segment.

There are other functions such as guards, which allow the user to define the enabling condition of a transition as a function of tokens in various places, or functions that allow the user to define arrays of places and transitions or functions that pertain to sensitivity analysis. Assert Segment The assert function allows the evaluation of a logical condition on a marking of the SPN. For example, the assert definition:

f

assex% 0 I if tmark(“p2”3+mark(‘~p3”) ret= (RESJRROR) ; else returnCRES,NOERR); It

> != 4 I I enabledC’%ii”)

&& enabledf”t7”)3

J

will stop the execution in a marking where the sum of the number of tokens in places p2 and p3 is not 4, or where t 11 and t7 are both enabled. Ac-init and AC-reach Segment The function ac-init is called just before starting the reachability graph construction. It can be used to output data about

SPNP

583

the SPN in the “. out” file. This is especially useful when the number of places or transitions is defined at run time (otherwise it is merely a summary of the CSPL file). The function ac-reach is called after the reachability graph construction is completed. It can be used to output data about the reachability graph in the “. out” file. The function ac-final is called after the solution of the CTMC has been completed, to carry out the computation and printing of user-requested outputs. For example, the following function: AC-final Segment

writes in the “ . out” file for each place the probability that it is not empty and its average number of tokens, and for each transition the probability that it is enabled and its average throughput, respectively. The derivatives of all the preceding standard measures with respect to a set of previously chosen parameters are computed and printed. In the end it writes the message goodbye. Before applying the ac-f inal function, additional functions often have to be provided. This can be done, for instance, using the construct reward-type as it is shown in the following example: reward-type reward-type reward-type l

.

epl(l c return(mark(“p1”)) ep30 I return(mark(“p3”)) ep7 (1 < return(mark(“p7”)

; 3 ; 3 >; 3

.

ac,f inaL (1 -C x = expected(epl)*expectedCsp7)+expected(ep3)*1.2; printf (“%f I’ ,x1 ;

3

1

Example Using the example of Fig. 12.9, we show that the application of SPNP to analyse systems that can be modeled by an SPN is a straightforward procedure. The inhibitor arcs avoid a capacity overflow in places pl and p2. The CSPL file for this SPN is shown in Fig. 12.10. A short description of this specification follows:

Line 4-6: The reachability graph, the CTMC, and the state probabilities the CTMC are to be printed.

of

Line 10-22: The places, the initial marking of the places, the transitions, their firing rates, and the input, output, and inhibitor arcs are defined. Line 26-29: Check whether a capacity overflow in either place pl or place p2 has occurred.

584

PERFORMANCE ANALYSIS TOOLS

Pl

P2 fig. 12.9

Fig. 12.10

Petri net example.

CSPL file for the SPN in Firr. 12.9.

SPNP

Line 33-37: Data about the SPN and the reachability

585

graph of the SPN are

to be printed.

Line 40: The number

is the reward

Line 41: The number

is the reward

of tokens in place pl in each marking rate sl in that marking.

of tokens in place p2 in each marking rate s2 in that marking.

Line 42-49: Self-explanatory. Line 52: Data about the CTMC

and its solution

are to be printed.

Line 53-54: The mean number of tokens in the places pl and p2 are to be computed

and printed.

Line 55-57: The probabilities computed

12.2.3

that the places pl and p2 are empty

are to be

and printed.

iSPN

Input to SPNP is specified using CSPL, but iSPN (integrated environment for modeling using stochastic Petri nets) removes this burden from the user by providing an interface for graphical representation of the model. The use of Tcl/Tk [Welc95] in designing iSPN makes this application portable to all platforms [HWFT97]. The major components of the iSPN interface (see Fig. 12.11) are a Petri net editor, which allows graphical input of the stochastic Petri net, and an extensive collection of visualization routines to display results of SPNP and aid in debugging. Each module in iSPN is briefly described in the following:

Input

Data: iSPN provides a higher level input format to CSPL, which provides great flexibility to users. iSPN is capable of executing SPNP with two different file formats: Files created directly using the CSPL and files created using iSPN’s Petri net editor. The Petri net editor (see Fig. 12.12), the software module of iSPN that allows users to graphically design the input models, introduces another way of programming SPN P: The user can draw the SPN model and establish all the necessary additional functions (i.e., rewards rates, guards, etc.) through a common environment. The Petri net editor provides several characteristics normally available only in sophisticated two-dimensional graphical editors and a lot of features designed specifically for the SP N P environment.

iSPN also provides a textural interface, which is necessary if we wish to accommodate several categories of users. Beginners may feel more comfortable using the Petri net editor, whereas experienced SPNP users

586

PERFORMANTE

ANA1 YSIS TOOI 5

Modeling Environments

Manager

Y Fig. 12.11 Main software modules for iSPN.

may wish to input their models using CSPL directly. Even if the textual input is the option of choice, many facilities are offered through the integrated environment. In both cases, the “SPNP control” module provides everything a user needs to run and control the execution of SP N P without having to switch back and forth among different environments. Output Data: The main goal of most GUIs is only to facilitate the creation of input data for its underlying package. Usually, the return communication between the software output and the GUI is neglected. One of the advantages of iSPN is the incorporation of displaying SPNP results in the GUI application. iSPN’s own graphing capability allows the results of experiments to be graphically displayed in the same environment (see Fig. 12.13). Different combinations of input data may be compared against each other on one plot or viewed simultaneously. The graphical output format is created in such a way that it may be viewed by other visualization packages such as gnuplot of xvgr.

SPNP

fig. 12.12

The Petri net editor.

Fig. 12.13

The iSPN output page.

587

588

PERFORMANCE ANALYSIS TOOLS

12.3

MOSES

In this section the CTMC-based tool MOSES (modeling, specification, and evaluation system), developed at the University of Erlangen, is introduced. MOSES is based on the system description language MOSEL (modeling specification, and evaluation language) [BoHe96]. The core of MOSEL consists of several constructs to specify the possible state and the state transitions of the CTMC to be analyzed. The basic way that MOSES computes the performance measures is shown in Fig. 12.14.

Fig. 12.14

Different steps when computing the performance measures with MOSES.

MOSES gets as input the system description (in MOSEL) and develops the underlying generator matrix Q. Five solution methods for the CTMC are provided (Grassmann (see Section 3.3.2), Jacobi (see Section 3.4.3), Gauss-Seidel (see Section 3.4.4), multilevel method (see [Hort96]), and uniformization (see Section 5.1.4)). From the state-probability vector, the system performance measures of interest are computed. In the following section we provide a brief overview of the model description language M OSEL. 12.3.1

The Model

Description

Language

MOSEL

MOSEL consists of a series of segments, described in the following: The DECLARATION

Part:

This method of definition is similar to the corresponding C constructs except that it only allows numerical values (integer and floating point) and no expressions or other types of assignments.

#define

#enum This construct allows the definition of a set of constants. If we do not assign any integer value to a constant name in the enumeration list, then this constant gets a default integer value. For example, the following

MOSES

589

two enum declarations are both identical: enumcpu,stal;e (idle enum cpu,stats

= 0, user = 1, kernel = 2, driver ~idle,user,kern~l,drives);

= 3);

This defines how a state vector looks The VECTOR Description Part: and which state vectors are prohibited (NOT Part). (NODES Part) In this part of the specification the components of the state NODESPart vector are specified. For each component of the state vector, its name and capacity (since we consider finite CTMCs) must be given. As an example, consider: ; /a Range: Q,,K */ NODEnade,lCKI NUDEcpuLcpu,stateI; /* Range: 0+.3 */

For the definition of cpu-state, see the example in the #enum construct given previously. There is also the possibility for defining subnodes within a NODE declaration. For instance: NDDENICK; Nlifi];

N22[1]]

/* Node Nl could model a M/E2/1 queue *

Subnodes can, for example, be used for components with Erlang distributed service times. START Part This optional part specifies a valid start state for the transient analysis. NOT Part This construct is used to specify the prohibited system states. If we have, for example, a closed queueing network, then the sum of jobs over all stations in the network cannot be different from K (the overall number of jobs in the system). The RULES Part: This is the most important segment in the system because the RULE constructs are used to specify the state transitions. Each rule consists of a global part and, optionally, a local part. Consider, for example, the following two rules where the first rule consists only of a global part while the second one has also a local part, consisting of two local rules:

The first rule specifies a transition from node-l to node-2 with transition rate mu-l. Since we do not specify a routing probability, 1.0 is chosen as default value. It should be noted here that instead of directly specifying the state transitions of the underlying CTMC (although it can be done), service

590

PERFORMANCE ANALYSIS TOOLS

rates of stations and routing probabilities of the network are specified. This compact method of specification is much easier and less error prone. The second rule consists of two local rules. The global part states that transitions begin at node-l and all take place with transition rate mu-l. These transitions occur only if statenode-l == up. In the local part we define the transition It is even possible to specify local conditions for targets and probabilities. each local rule. In this case the global as well as local conditions have to be satisfied for the transition to occur.

The RESULT CTMC ed.

Part

In this part the basic performance measures of the from the steady-state probabilities are requested to be print-

computed

Some parts of a specification happen to be very similar, Loops in MOSEL differing, for example, only in one index. For situations like these, MOSEL offers the loop construct. Consider the following example specification:

It is very cumbersome and error prone to repeat every single specification since they differ from each other only in the index. These three lines of code can be shortened using the loop construct: <2..4> FROMnode-# TO node-1 W mu-# P p#l ; As we can see, the loop indices are specified 12.3.2

within

the angled brackets.

Examples

12.3.2.1 CentraLServer Mode/ In the following the central-server model is specified. By using MOSEL, the queueing network model shown in Fig. 12.15 can be specified in many different ways. Here we present two different versions: a longer one that is quite straightforward (see Fig. 12.16) and a second one that is a much shorter specification and demonstrates the power of loops (Fig. 12.17). A short description of the lines in the specification of Fig. 12.16 follows:

Line 2: Definition Line 5-7: Definition

of the number

of jobs in the system.

of the state space.

Line 9: Definition

of the prohibited states in the system. sum of all jobs at all nodes must be I(.

Line 13-14: Specification other nodes.

of all possible state transitions

At any time, the

from node N1. to all

MOSES

fig. 12.15

Fig. 12.16

MOSEL implementation

Line 15-16: Specification node Nl.

591

The central-server model.

of the central-server model without loops.

of all possible transitions from nodes N2 and N3 to

Line 18-20: Specification of the utilization of a node using the UTIL construct (which computes the utilization of a node using the state probabilities). Line 22-24: Specification of the mean number of jobs at a node using the MEAN construct. Line 26-32: Definition of other interesting performance measures. As we can see, the use of loops considerably reduces the length of the specification. Here we show the usability of 12.3.2.2 Fault Tolerant Multiprocessor System MOSEL for the model of a fault tolerant multiprocessor system. Fig. 12.18 shows a multiprocessor system with a finite buffer capacity. It is assumed that the failure and repair of servers are mutually independent. The following characteristics also apply:

592

PERFORMANCE ANALYSIS TOOLS

Fig. 12.17

MOSEL implementation

Fig. 12.18

of the central-server system with loops.

An example of a multiprocessor

l

The system has a finite job queue of length

l

The arrival

l

The time to repair is exponentially

l

The time llmtbf.

l

If a server that is currently again on a free server.

l

If no server is currently available, repaired or becomes free.

l

Service times are exponentially

system.

L.

rate to this queue is lambda.

between

failures

distributed

is exponentially

working

with the rate l/mttr. distributed

with

the rate

on a job fails, then the job is started

then a job must wait until

distributed

a server is

with mean l/p.

The implementation in Fig. 12.19 shows one way to describe this model of the fault tolerant multiprocessor system. Descriptions of the lines follow:

Line 2-4: Definition

of the states space.

593

MOSES

/

1 2 3 4 5 6 7 8 Q 10 11 12 13 14 15 16 17 18 IQ 20 21 22 23 L24

Fig. 12.19

/* VECTORdascription NODEqueue[Ll NODEactiveC31 NODEworkingC31 NOT queue*active

‘\

part */

> L+3;

/* RULESpart FROME FROMworking FROME

*/ TO queue W lambda; W l/mtbf; TOE TO working W l/mttr;

FROMqueue

TO active

IF working-active

3 0;

<1..3> FROMactive TOE W #*mu IF active==# AND work$ng>*#; <1.,2> FROMactive TOE W #*mu IF active># AND working-#; /* RESULTS part */ RESULTA = MEANactive; RESULTrho = A/3; RESULTIF (working > 0) p-working +* PROB; RESULTDIST queue; RESULTq = MEANqueue; RESULT>) throughput = 3*rho*mu*p,working;

MOSEL implementation

J

of the fault tolerant multiprocessor system

Line 6: Definition

of the prohibited states in the system. The sum of the active jobs and the jobs in the queue shall not exceed L+3.

Line 9: Specification

of the arrival for FROM External).

of jobs in the system.

(FROME: short form

Line 10-11: Specification

and the repair of a processor.

Line 13: Specification

from the queue to a processor.

of the failure short form for TO External). of the transition

Line 15-16: Specification

(TOE:

of the possible transitions from the processor to external. If a processor fails while the job is being processed, the job is started on a free server, if available, or has to wait.

Line 19: Request the mean number of active processors. Line 20: Request the utilization Line 21: Request the probability

of the processors. that at least one processor

is working.

Line 22: Request the pmf of the number of jobs in the queue. Line 23: Request the mean queue length. Line 24: Request the throughput.

594

PERFORMANCE ANALYSIS TOOLS

12.4

SHARPE

SHARPE (symbolic hierarchical automated reliability performance evaluator) tool was originally developed in 1986 by Sahner and Trivedi at Duke University [STP96]. It is implemented in C and runs on virtually all platforms. The advantage of SHARPE is that it is not restricted to one model type as PEPSY or SPNP. It offers seven different model types including product-form queueing networks, stochastic Petri nets, CTMCs, task precedence graphs and fault trees. The user can choose the model type that is most convenient for the problem. The models can also be hierarchical, which means the output of a submodel can be used as the input for another submodel. Several graphical user interfaces for SHARPE are now available but the original version used textual, line-oriented ASCII code. An interactive or batch oriented input is possible. The syntax for both cases is the same. The possibilities that SHARPE offers depend on the chosen model type. If the user wishes to analyze a product-form queueing network, the commands for the performance measures such as throughput, utilization, mean response time at a station, and mean queue length are provided by SHARPE. MVA is used to solve product-form queueing networks. For a continuous-time Markov chain, reward rates and initial state probability vector can be specified. Steadystate, transient, and cumulative transient measures can be computed. The tool offers two steady-state solution methods. At first it starts with the SOR method and after a certain period of time, it prints the number of iteration steps and the tolerance if no solution is found. In this case the user is asked whether he or she wants to go on with the SOR method or switch to Gauss Seidel. Three different transient CTMC solution methods are available. It is not possible in this context to describe the full power of SHARPE. Instead of describing the syntax and semantics in detail, we give examples that use the most important model types to show the reader that SHARPE is very powerful and also easy to use. SHARPE has been installed at more than 220 sites. For more detailed information the reader can consult [STP96]. 12.4.1

Central-Server

Queueing

Network

As the first example we consider a central-server queueing network (Fig. 12.20) consisting of a CPU and two disks with the following input parameters: /Lo = 1000/20, Pll

=

0.1,

p1 = 1000/30, p12 = 0.667,

p2 = 1000/42.9, p13 = 0.233,

P2l = P31 = 1.

At each node the scheduling discipline is FCFS and the service times are exponentially distributed. Figure 12.21 shows a SHARPE input file for this

SHARPE

595

Disk 2 Fig. 12.20

\

f

I

A central-server rnodcl.

1

* central-server

2 3 4 5 6 7 8 9 10 11 12 13 14 16 16

pfqn csm(jobs) cpu diski ~12 cpu disk2 ~13 disk1 cpu 2 disk2 cpu 1 end * servers cpu fcfs 1000/20 diski fcfs 1000/30 disk2 fcfs ~1000/42.918 end * numberof jobs chain1 jobs end Fig. 12.21

mod&

17 18 19 20 21 22 23 24 25 26 27 28 29 30

bind ~12 0.667 p13 0,223 end

Soap i,Z,lG,2

expr expr expr: expr end

tputCcsm,cpu;i) util(csm,cpu;i) qlength(csm,cputi) rtime(csm,cpu; i>

end /

SHARPE Input for central-server network.

model. Lines with comments begin with an asterisk. The description of the SHARPE specification is given as follows: Line 3: Specifies the model type pf qn (= product-form model named csm, and model parameter jobs.

queueing network),

Line 4-8: Specification of the routing probabilities. Line 9-13: Specification of the nodes and the service parameters of the nodes (name, queueing discipline, service rate). Line 15-16: Gives the number of jobs in the network. Line 18-21: The routing probabilities

are bound to specific values.

Line 23-28: Requests the computation and printing of the throughput, the utilization, the mean queue length, and the mean response time at the cpu for the number of jobs: 2, 4, 6, 8, and 10. The output produced by SHARPE in this case is shown in Fig. 12.22. We see that the throughput grows from 29 to 45, the utilization from 0.59 to 0.90, the mean queue length from 0.82 to 4.60, and the mean response time from 0.03 to 0.10, as the number of jobs increases from 2 to 10.

596

PERFORMANCE ANALYSIS TOOLS

f

L

2.9406&01 5.88lle-01 8.23318-01 2.7998e-02

i=8.000000 tput(csm,cpu;i): utilCcsm,cpu;i): qlength(csm,cpu;i): rtime(csm,cpu;i):

4,37E;3e+Ol 8.75066-01 3,6209e+OO 8.2768e-02

i=4.000000 tputCcsm,cpu;i): util(csm,cpu;i): qlength(csm,cpu;i): rtime(csm,cpu;i):

3.7976e+Ol 7.5962e-0 1.7202e+OO 4.6298e-02

i=10.000000 tput(csm,cpu;iI: util(csm,cpu;i): qlength(csm,cpu;iI: rtime(csm,cpu;i):

4*4992e+Ol 8.99836-01 4.59556+00 1.02146-01

i=6.000000 tput(csm,cpu;i): util(csm,cpu;i): qlength(csm,cpu;i): rtime(csm,cpu;i):

4.1733e+Ol 8.3466e-01 2.659ie+OO 6,3717e-02

fig. 12.22

12.4.2

\

i=2.000000 tput(csm,cpu;i): util(csm,cpu;i): qlength(csm,cpu;i): rtime(csm,cpu;i):

M/M/m/K

/

SHARPE output for the central-scrvcr model.

System

As a simple example for a CTMC model, we consider an M/M/5/100 queueing system. The state diagram for this CTMC is given in Fig. 12.23. It is of birth-

Fig. 12.23

Markov chain for the M/M/5/100

system.

death type and irreducible. Each state name gives the number of jobs in the system. The SHARPE input file is shown in Fig. 12.24. In the first two lines the values of the arrival and service rates are bound. Then the model type (markov = CTMC) and the name (mm5) of the model are given. Subsequent lines specify the transitions between the state together with their transition rates. Then five variables are defined. Pidle is the steady-state probability of being in state 0. This is the probability that there are no jobs being served, i.e., that the station is idle. Pfull is the steady-state probability of being in state 100. This is the probability that the queue is full. Lre j ect is the rate at which jobs are rejected. Mqueue is the mean number of jobs in the system. This can be calculated by the built-in function sum. Mresp is the mean response time of accepted jobs computed using Little’s theorem as Mqueue/(X-Lreject), expr expression prints the value of the expression in the output file (Fig. 12.25). Note that a recent extension to SHARPE includes a loop specification in the definition of a CTMC that can be used to make concise specification of a

SHARPE

597

structured CTMC such as the one just described. For an example of the use of this feature see Section 13.3.2.

Fig. 12.25

12.4.3

M/M/l/K

Output file for the M/M/5/100

System

with

Server

Failure

system.

and Repair

Now we extend the M/M/l/K queueing model by allowing for the possibility that a server could fail and could be repaired. Let the job arrival rate be X and the job service rate be p. The processor failure rate is y and the processor repair rate is 7. This system can be modeled using an irreducible CTMC, as shown in Fig. 12.26 for the case where m = 1 (one server) and IY = 10 (the maximum number of jobs in the server and queue is 10). Each state is named with a two-digit number ij where i E (0, 1, . . . ,9, a} is the number of jobs in the system (a is 10 in hexadecimal) and j E (0, l} denotes the number of operational processors. SHARPE input and output files for this example are shown in Figs. 12.27 and 12.28. The probability that the system + prob(mmlk,Ol), and the rate at which is idle is given by prob(mmlk,OO) jobs are rejected because the system is full is given by X (prob(mmlk, d) + prob(mmlk,

al)).

The generalized stochastic Petri net (GSPN) in Fig. 12.29 is equivalent to the CTMC model and will let us vary the value of K without changing the model structure. We use this example to show how to handle GSPNs

598

PERFORMANCE ANALYSIS TOOLS

Fig. 12.26

f

CTMC for an M/M/1/10

markov mmlk 01 11 LAM 1% ot Mu

11 10 CAM 10 ii TAU

system with server failure

and repair.

\

30 40 LAM

21 31 LAM

30 31 TAU

40 50 LAM 60 70 LAM ‘70 60 LAM 80 90 LAM 90 a0 LAM

31 21 Mu 31 41 LAM 41 31 Mu

41 40 GAM

end

40 41 TAU

41 62 LAM

50 51 TAU

bind LAM 1

61 41 Mu

61 60 GAM

MU 2

51 61 LAM 61 51 MU 62 71 LAM

60 61 TAU

GAM O.UUOl

70 71 TAU

71 70 GAM

TAU 0.1

71 61 MU 72 81 LAM

81 80 GAM

var Pidle

probCmmik,OO)+~rob~mlk,Ol)

80 81 TAU

var Pfull

prob~mmik,a0)+grob~mlk,al)

91 90 al a0 00 10

var Lreject exps Pidle

11 21 LAM 21 11 Mu

81 81 91 92 al 01 \, 00

71 91 81 al 91 00 01

Fig. 12.27

MU LAM MU LAM

Mu GAM TAU

21 10 GAM 20 21 TAU 31 30 GAM

61 60 GAM

90 91 a0 al IO 20

GAM TAU GAM TAU LAM LAM

end

LAM*PFULL

expr Lre j ccc end

20 30 LAM

/

SHARPE input for the M/M/1/10

system with server failure

and repair.

using SHARPE. The loop in the upper part of the GSPN is a representation of an M/M/l/K queue. The lower loop models a server that can fail and be repaired. The inhibitor arc from place server-down to transition service shows that customers cannot be served while the server is not functioning. The number within each place is the initial number of tokens in the place. All of the transitions are timed, and their firing rates are shown below the transitions. The input file for this model is shown in Fig. 12.30, and the description of this file follows:

Line l-7:

The input

parameters

are bound to specific values.

Line 9: Model type and name of the model. Line 11-15: Places and initial

numbers

of the tokens.

SHA RPE

fig. 12.28

SHARPE output

Fig. 12.29

for the M/M/1/10

system with server failure

GSPN model for queue with server failure

599

and repair.

and repair.

Line 17-21: Timed transitions. Line 23-27: Arcs from places to transitions and arc multiplicities. Line 29-33: Arcs from transitions to places and arc multiplicities. Line 35-36: Inhibitor

arcs and their multiplicities.

Line 38: Define a variable for the probability Line 40: Define a variable for the probability server.

that the server is idle. that a job is rejected at the

Line 42: Define a variable for the rejection rate. Line 44: Define a variable for the average queue length at server. Line 46: Define a variable for the throughput Line 48: Define a variable for the utilization

of server. of server.

Line 50-54: Values requested to be printed (Fig. 12.31). This GSPN is irreducible. For GSPNs that are non-irreducible, SHARPE can compute the expected number of tokens in a place at a particular time t, the probability that a place is empty at time t, the throughput and utilization of a transition at time t, the time-averaged number of tokens in a place during the interval (O,t), and the time-averaged throughput of a transition during (0,t). See [STP96] for the syntax for these functions. Other model types

PERFORMANCE ANALYSIS TOOLS

600

’

1. 2 3 4 6 6 7 8

9

10 11 12 13 14 16 26 17 18 19 20 21 22 23 24 25 26 27

bind Lambda 1.0 mu 2.0

28 29 30 31 32 33 34 36 36 37 38 39 40 41 42 43 44 46 46 47 48 49 60 51 52 53 64

gamma 0.0001.

tau 0.1 K 10 end gspn mmlk-fail * places jobsource K queue 0 servexup 1. serverdown 0 end * transitions job-arrival ind Lambda service ind mu faiLuxe ind gamma repair ind tau end * arcs, places-transitions jobsouxce job-arrival 1, queue service I serverup failure 1 serverdown repair 1 end

Fig. 12.30

Input

* axes, transitions-places job-arrival queue I service jobsource I faiLwe servexdown 1 repaix sexverup 1, and * inhibitox arcs serverdown service 1 end

I

vax Pidle prempty(mmik-fail,q

cue> u vax Pxeject prempty(mmlk-fail/jobsource) var Lrejeet prempty(mmik-failrjobsource) var avquelength etok(mmlk-fail,queue$ vax thxuput tput(mmlk-fail.,service) var utiY.ization expx expr exps expr end

uitl(mmlk-fail,service)

PicUe Lreject, Preject avquelength thruput, utilization

for GSPN model of t,he system with server failure

avquelength:

1

I and repair.

1,0028e+OO

9.9@04e-01 utilization:

Fig. 12.31

Output

for the GSPN model.

provided by SHARPE are multiple chain product-form queueing networks, semi-Markov chains, reliability block diagrams, fault trees, reliability graphs, and series-parallel graphs. For details see [STP96].

12.5

CHARACTERISTICS

OF SOME TOOLS

In Table 12.1 other important tools are listed together with their main features. There are pure queueing network tools such as QNAP2 or PEPSY, pure Petri net tools such as SPNP or PANDA, and tools that have a more general input and are therefore more flexible such as SHARPE or MOSES. Many of them provide a GUI and/or a textual input language. The GUI is very convenient for getting accustomed to the tools, but the experienced user often prefers the use of a textual input language.

Table 12.1 Tool

Location

Reference QN

RESQ QNAP2 MAOS PEPSY HIT

Performance

Modeltype SPN other

DES

Evaluation

Solution Method pqfn npqfn m-tr

*

(*)

* ILMAOS

1 -

i *

XPEPSY HITGRAPHIC

HISLANG

* * * * * *

iSPN

CSPL

1 -

* * * * * *

-

-

-

*

*

APNN

*

* -

* -

* * * -

* * * * *

GSHARPE

* MOSEL

* *

-

-

*

*

*

-

-

*

*

TU Miinchen Univ. Erlangen Univ. Dortmund

[FeJoSO] [Kirs94] [BMW891

* * *

-

1 -

* * *

* * *

i -

SPNP GreatSPN TIMENET DSPNexpress Ultra SAN PANDA

Duke Univ. Univ. Torino TU Berlin TU Berlin Univ. Illinois Univ. Erlangen

[CFMT94] [ABC+951 [GKZH94] [Lind94] [SOQW95] [AlDa97]

-

* * * * * *

-

* * *

-

-

i *

1 -

QPN

Univ.

[BaKe94]

*

*

-

-

SHARPE MOSES MARCA MACOM DNAmaca

Duke Univ. Univ. Erlangen NC State Univ. Univ. Dortmund Univ. Cape Town

[STP96] [BoHe96] [StewSO] [ScMiiSO] [KnKr96]

* * * * *

* * *

* * * *

(i) -

solver

steady-state

CTMC

solver

transient,

m-ss:

CTMC

Hierarchical

*

*

[VePo85]

m-tr:

Language

-

[SaMa85]

SIMULOG

Notes:

GUI m-ss

*

IBM

Dortmund

Tools

-

-

-

* * * * *

(*)

*

(Menue)

*

(I)

2

*

XMARCA *

(“SE*N”M)

1

C++-constructs

-

2i % 4

5 i a % $

m

13 Applications

This chapter considers several large applications. The set of applications organized into three sections. In Section 13.1, we present case studies queueing network applications. In Section 13.2 we present case studies Markov chains and stochastic Petri nets. In Section 13.3, case studies hierarchical models are presented.

13.1

CASE STUDIES

OF QUEUEING

is of of of

NETWORKS

Five different case studies are presented in this section. These range from multiprocessor system model, several networking applications one operating system model and a flexible production system model. 13.1.1

Multiprocessor

Systems

Models of tightly coupled multiprocessor systems will be discussed first followed by models of loosely coupled systems. 13.1,l.l Tightly Coupled Systems Consider a tightly coupled multiprocessor system with caches at each processor and a common memory, connected to the processors via a common bus (see Fig. 13.1). The system consists of m processors and a common memory with n memory modules. A processor sends a request via the common bus to one of the n memory modules when a cache miss occurs, whereupon the requested data is loaded into the cache via

603

604

APPLICATIONS

fig. 13.1

PTl

Processor n

72= 1,...,5

cn

Cache n

n=

iVIAl,

Memory Module n n = 1,. . . ,4

A multiprocessor

1,...,5

system with caches, common

the common bus. Figure 13.2 shows a product-form of such a multiprocessor system.

memory,

and common

bus.

queueing network model

Replies to Memory Requests

Replies to Memory Requests fig. 13.2

Queueing

network

Memory Modules

model of the multiprocessor

system shown in Fig. 13.1.

The m processors are modeled by an IS node and the bus and the memory modules by single server nodes. The number of requests in the system is m since a processor is either working or waiting until a memory request is finished. Using this model we can calculate the utilization of the bus or the mean response time of a memory request from a processor. The mean time between two cache misses is modeled by the mean thinking time of the IS node. Other parameters that we need are the mean bus service time, the mean memory request time, and, finally, pi, the probability of a request to memory module i. In the absence of additional information, we assume pi = l/n

CASE STUDIES OF QUEUEING NETWORKS

605

(i = 1,2,... , n). Note that for each cache miss, there are two service requests to the bus. It is for this reason that with probability 0.5, a completed bus request returns to the processor station. Similarly, the probability of visiting memory module i subsequent to the completion of a bus request is pi/2. We assume that the service times for the two types of bus requests have the same mean. This assumption can be easily relaxed. If we have explicit values for these parameters, then we can calculate interesting performance measures such as bus utilization and mean response time as functions of the number of processors. Another interesting measure is the increase in the mean response time assuming a fixed number of processors and memory modules, while changing the mean bus service time and the mean time between two cache misses. In Fig. 13.3 the mean response time is shown as a function of the mean bus service time and the mean time between two cache misses [BolcSl].

Mean

R.espont Time

4ce Time

Mean Time Between Cache Misses fig. 13.3 Mean response time as a function time between cache misses (m = 5, n = 4).

5o

of the bus service time,

and the mean

We can see that there is a wide area where the increase in the mean response o compared to the case with no waiting time at the bus time is tolerable (1501) or a memory queue (this is the minimum value of the mean response time). On the other hand, there is also a wide area with an intolerable increase in the mean response time. Thus the analysis of such a queueing network model can help the system designer to choose parameter values in a tolerable area. Problem 13.1 Verify the results shown in Fig. 13.3 by hand computation and using any software package available (e.g., SHARPE or PEPSY).

606

APPLICATIONS

Problem

13.2

Improve the model described in Section 13.1.1.1 by considering different mean service times for bus requests from processor to memory and from memory to cache (Hint: Use a multiclass queueing network model). In a loosely coupled multiprocessor system, the processors have only local memory and local I/O devices, and communicate via an interconnection network by exchanging messages. A simple queueing network model of such a system is shown in Fig. 13.4. The mean 13.1.1.2

Loosely

Coupled

Systems

Network

Processors

Fig. 13.4

A simple queueing

network

model of a loosely coupled

system.

service time at the processors is the mean time between the messages, and the mean service time at the network node N is the mean message delay at the network. The routing probability pi is the probability that a message is directed to processor i. Arriving

/ Departing

Jobs

Replies to I/O Requests

Processors Replies to Memory Requests

Fig. 13.5

A complex

queueing

network

I/O Processors model of a loosely coupled system.

A more complex and detailed model of a loosely coupled multiprocessor system is shown in Fig. 13.5 [MAD94]. Here we assume that we have n separate The computing processors processors. I/O processors and m “computing” send I/O requests via the network N to the I/O processors and get the replies to these requests also via the network N. We assume that a job that begins at a computing processor is also completed at this processor and that the processors are not multiprogrammed and are heavily loaded (for each job that leaves the system after being processed by a processor, a new job arrives immediate-

CASE STUDIES OF QUEUEING NETWORKS

607

ly at the processor). This system can be modeled by a closed product-form queueing network with m job classes (one for each computing processor) with the population of each class equal to 1 [MAD94]. As an example we consider a loosely coupled multiprocessor system with eight computing processors and n = 2,3,4 I/O processors. The mean computing processor service time is 30 msec, the mean I/O time is 50 msec, and the mean message delay at the network is 1 msec. The probability that a job leaves the system after it has been processed at a computing processor is p&ne = 0.05. we assume that pi = l/n for all i. Some interesting results from this model are listed in Table 13.1. Table 13.1 Performance measures for the loosely coupled multiprocessor different, numbers of I/O processors 2

Number of I/O Processors

4.15 set 1.93 set- l 0.145 0.070 0.867

mean response time throughput Pcomputingprocessor Pnetwork PI/Oprocessor

3 3.18 set 2.51 set-l 0.189 0.090 0.754

system for

4 2.719 set 2.944 set-l 0.220 0.106 0.662

Problem 13.3 Verify the results shown in Table 13.1 by hand computation and using any software package available (e.g., PEPSY or SHARPE). Problem

13.4 Extend the complex model described in Section 13.1.1.2 so as to allow distinct mean network service times for request from a computing processor to I/O processor and vice versa. 13.1.2

Client

Server

Systems

A client server system consists of client and server processes and some method of interprocess communication. Usually the client and the server processes are executing on different machines and are connected by a LAN. The client interacts with the user, generates requests for a server, transmits the request to the server, receives the results from the server, and presents the results to the user. The server responds to requests from the clients and controls access to resources such as file systems, databases, wide area networks, or printers. As an example we consider a client server system with a fixed number m of client workstations that are connected by an Ethernet network to a database server. The server consists of a single disk (node number 4) and a single CPU (node number 3). This leads to a closed product-form queueing network model shown in Fig. 13.6. The client workstations are modeled as an IS node (node number 1) [MAD94], and the number of jobs in the closed system is equal to the number of workstations m. The Ethernet network (carrier sense multiple access with

608

APPLICATIONS

Workstations

fig.

13.6 Closed

qucueing

network model of a client server system.

collision detection, or CSMA/CD, network) can be modeled as a server (node number 2) with the load-dependent service rate [LZGS84, MAD94, HLM96]:

(

-

y$ * 3

PIlet (W =

i

+ s * c(l))-l,

~.~+S*C(t+l))-l, (

k = 1, (13.1) k>l,

where C(k) = (1- A(k))/A(k) is tkre average number of collisions per request is the probability of a successful transmission and Ic and A(k) = (1 - l/k)'-' the number of workstations that desire the use of the network. Other parameters are described in Table 13.2 and shown in Fig. 13.6. We Table 13.2

Parameters for the client server example

Parameter

NP B S

GJ

Description Average number of packets generated per request Network bandwidth in bits per second Slot duration (i.e., time for collision detection) Average packet length in bits

compute the throughput X and other interesting performance measures using the load-dependent MVA (see Section 8.2). As an example we use the parameters from Table 13.3 and determine the throughput as a function of the number of client workstations m (see Fig. 13.7) Problem 13.5 Verify the results shown in Fig. 13.7 by hand computation and using an available modeling package such as SHARPE or PEPSY.

13.1.3

Communication

Systems

13.1.3.1 Description of the System As an example of a more complex queueing network and the performance evaluation of communication systems, we consider the LAN of a medium size enterprise [Ehre96].

CASE STUDIES OF QUEUEING NETWORKS

Fig. 13.7

” 20

4b

Throughput

as a function of the number of workstations.

Table 13.3 Numerical server example Np = 7 B = 10 Mb/ set S = 51.2 psec z, = 1518 bits pa1 = 0.5 P12 = 1 p23 = 0.5

m 0

6b m

parameter

8b

160

values for the client-

pl = pc~ = O.l/ set CL2 = pNet(k) p3 = pcpu

= 16.7/set p~q= PDisk = 18.5/set p32 = 0.5 P43 = p34 = 0.5

1

Bridges in the Computer Center Bridges in the Other Buildings WAN Router LAN Analyzer

@ Server •i Computer

Fig. 13.8

The FDDI backbone with 13 stations.

609

610

APPLICATIONS

The LAN connects several buildings that are close together and it is divided into several network sections. In each network section the devices of a building are connected to each other. Three other sections are located in one building and are used to connect the servers to the LAN. As a backbone for the sections, a fiber distributed data interface (FDDI) rin g is used. Eleven bridges for the sections, a WAN router, and a LAN analyzer that measures the utilization of the LAN constitute the stations of this ring. Figure 13.8 shows the FDDI ring with the stations schematically, wherein the structure of two stations is shown in more detail. The typical structure of a section in which all computers within a building are connected is shown in Fig. 13.9. The building has four floors. The computers on the second and third floor are connected to different segments, and the computers on the fourth and fifth floor are connected to the same segment. These three segments are connected to each other and to the FDDI ring via a multiport bridge. The computers are connected to a stack of four hubs that

5th Floor

4th Floor

Cheapernet I

I

I 3rd Floor

Chcapernet Optical Fiber

I

1 ---p-&

2nd Floor (Ground Floor) -x Hubs FDDI

Fig. 13.9

A section of a communication

Ring

system with three segments.

are connected to each other via Cheapernet, and there is a connection of the hubs to the multiport bridge via optical fibers. The CSMA/CD principle is used with a transmission rate of 10 Mb/set which is equal for all segments. Each segment is a different collision domain.

CASE STUDIES OF QUEUEING NETWORKS

611

The nodes of the queueing network model 13.1.3.2 Queueing Network Mode/ of the LAN are the FDDI ring, the Ethernets that connect the devices in the segment, the computers, the servers in the computer center, and the bridges. The FDDI ring can be considered as a server that serves the stations in the ring (12 stations without the LAN analyzer) in a fixed sequence. A station (or section) sends a request to another station via the FDDI ring, and the reply to the request is sent back to the station again via the FDDI ring. Thus we obtain a closed queueing network model for the LAN as shown in Fig. 13.10 with a simplified representation of the individual sections. The FDDI ring is modeled by a single server node. The LAN analyzer does not influence the other sections and for this reason it does not appear in the model. The WAN router can be modeled as a single server node and the WAN itself as an IS node (see Fig. 13.11). There are arrivals at the WAN router from the WAN

FDDI

P4

P5

P3

PG

P2

P7

Ring

*1 4-Ixt-+ +Ict-3 Sec&yye;swith

ps qT-+* pg q--q--@ Pl

WAN

pl”

+55--j-*

p11 +i-j--* p12 +-ii+--~ 8 Sections

with

Computers

Fig. 13.10 Closed queueing network model of the LAN with a simplified representation of the individual sections. and from the ring. The WAN router sends requests to the ring and to the WAN as well. A similar situation occurs at all the bridges/routers. In the sections, the segments are connected to the bridge via Ethernet. The Ethernet segments are modeled by a multiple server with only one queue where the number of servers is the number of segments in the section. Because of the CSMA/CD strategy, the frames are either transferred successfully to the computers of the sections to the bridge or in the case of a collision, sent back to the Ethernet queue with collision probability qi . Each computer sends requests to a server via the LAN and waits for the reply before it sends another request. Therefore the number of requests in the LAN equals the number of

612

APPLICATIONS From

To

From

To

A

The Ring

0.5 0.5

-oarWAN Router

WAN Fig. 13.11

Detailed

model of the WAN and the WAN router.

active computers, and the requests do not have to wait at the computers. For this reason the computers at a section can be modeled by an IS node. A queueing network model of a section of the LAN is shown in Fig. 13.12 and the queueing network model of a computer center section with the servers From

To

To

From

-IIllK+-

Bridge

42

:::

l--q, 2

l-q, 2

Computers Fig. 13.12

Detailed

model of a section of the LAN.

is shown in Fig. 13.13. Servers are modeled by a multiple server node with one queue where the number of servers at the node is equal to the number of real servers. The queue is necessary because there are more computers that can send requests to the servers than the number of available servers. The whole LAN model consists of 1 node for the FDDI ring, 12 nodes for the bridges, 11 nodes for the Ethernet segments, 8 nodes for the computers, 3 nodes for the servers, and 1 node for the WAN. As we will allow non-exponential service time distributions, it is a non-product-form network. As parameters we need the total number of 13.1.3.3 Model Parameters requests, the routing probabilities p,, the mean service times l/pi, and the

CASE STUDIES OF QUEUEING NETWORKS To

From

613

To

From

Server ”

Fig. 13.13

Detailed

model of a section with servers.

coefficient of variation c, of the service times of the nodes. Furthermore, we need the number of service units m; for the multiple server nodes. The total number of requests in the network is the mean number of active computers K = 170 (measured value). We estimate the routing probabilities pi from the FDDI ring to the bridges by measuring the amount of data transferred from the ring to the bridges, which can be measured at the bridges. This estimate was done based on measurements over a two-day period (see Table 13.4).

Table 13.4 Total data transferred and estimated routing probabilities

from FDDI

ring to the bridges for a two-day

period

p,

Section

1

2

3

4

5

6

7

8

9

10

11

12

Data/Mb Pzl%

2655 9.5

1690 6.2

2800 10

1652 5.9

2840 10.1

1500 5.4

3000 10.7

200 0.7

1940 6.9

1180 4.2

4360 14.8

4380 15.6

Notes: 1: WAN; 2,3,4: section with servers; 5, . . .,12: section with computers.

Table 13.5

Collision

qz

probabilities

Section

1

2

3

4

5

6

7

8

9

10

11

12

Qz/%

1

5

1

1

1

1

1

1

1

1

3

2

The other necessary routing probabilities are shown in Figs. 13.11, 13.12, and 13.13 and in Table 13.5. The number of servers m; of the multiple server nodes is given by the number of servers for the server nodes (Table 13.6) or by the number of segments for the Ethernet node (Table 13.7). To obtain the mean service times of the nodes, we use a LAN analyzer which can measure

614

APPLICATIONS

Table 13.6 Number of service units m, in server nodes

Table 13.7

Number

Section

2

3

4

mi

14

23

13

of service units nz; (= number

of segments)

in Ethernet

nodes

Section

2

3

4

5

6

7

8

9

10

11

12

ma

4

7

2

4

5

3

1

4

2

5

5

the interarrival times and the length of the frames transferred by the FDDI ring. The results of the measurement are shown in Tables 13.8 and 13.9. From Tab/e 13.8 Empirical pmf of the interarrival times of the frames. Interarrival

Time (ps)

15 5-20 20-82 82-328 329-1300 1300-5200

% 3.0 0.9 15.3 42.7 27.1 1.0

Tab/e 13.9 Empirical the frame length L

pmf of

Length (Bytes)

%

5 32 32-63 64-95 96-127 128-191 192-511 512-1023 1024-1526

11.1 10.0 33.0 7.9 6.8 4.1 5.8 21.3

these tables we obtain the mean values and the square coefficients of variation (Table 13.10) of the interarrival times and frame lengths. Given the measured throughput X = l/346 psec = 2890 per set, and the routing probabilities (see Table 13.4), the traffic equations are solved to produce the arrival rates of the individual sections (see Table 13.11). N ormally in a closed network, relative throughputs are computed from routing probabilities or visit ratios. However, in this case we have measured throughput at one node and hence the relative throughputs at each node are also the actual throughputs. The mean service time of the FDDI ring is given by the sum of the mean transfer time of a frame and the mean waiting time until the token arrives at a station [MSWSS]. We assume one-limited service, i.e., a station transmits at most one frame when the token arrives. An upper limit of the service time of the ring is the token rotation time. The utilization pi = Xi/p is the probability that a station wishes to transmit a frame, and 1 --pi is the probability that the token is transferred to the next station without transmitting a frame. In this case the transfer of the token can be considered as the transfer of a frame with

615

CASE STUDIES OF QUEUEING NETWORKS

Tab/e 13.10

Characteristic

values of the interarrival Interarrival

Length of the Frames

Times of the Frames

382.5 Byte 1.67

346 p set 1.48

Mean value Squared coefficient of variation

Tab/e 13.11

times and the frame length

Arrival

rates of frames at the stations

Station

1

2

3

4

5

6

7

8

9

10

11

12

A,

275

179

289

171

292

156

309

20

199

121

428

451

length

0. Accordingly,

the mean token rotation time ?& can be calculated as

follows: T,=U+R-‘*Z*&II~.

(13.2)

Here U denotes the free token rotation time (22psec), R denotes the transfer and 2; denotes the mean frame length (see Table 13.10). rate (100 Mb/set), With:

c

pi

=

Ipi x

(13.3)

-=-

P

P’

the approximation:

and Eq. (13.2), it follows that the service rate of the FDDI

‘=

R-XL U.R

ring is given by: (13.4)

*

With X = 2890/ set, z = 382.5 Bytes (see Table 13.10), we obtain service rate: p = 41435/set, and the mean service time at the FDDI

the mean

ring:

T, M 1 = 24 psec. I-L The variance

of the token rotation crgT = Re2 (p . var(L)

time T, is given by [MSW88]: + p. (1 - pep?)

z2)

,

(13.5)

616

APPLICATIONS

where o$ and z2 can be calculated using the values from Table 13.9, the values ofrpi are given in Table 13.4, and p = X/p = 0.070 is given by the values of X and p. Then we obtain the squared coefficient of variation:

The service time at each bridge is deterministic with the forwarding rate of 10,000 frames/set. And, of course, the coefficient of variation is 0 in this case. The service time of the Ethernet is given by the sum of the transfer time Tt and the delay time Td of a frame. To obtain the transfer time Tt we need the mean frame length, which we get from Table 13.9 given that the minimum frame size is 72 bytes in the CSMA/CD case. Thus, 54.1% of the frames have a size between 72 and 95 bytes and it follows: Leth = 395 bytes = 3160 bit,

ciplJL = 1.51.

Given a transfer rate of 10 Mb/set (see Table 13.12), we finally have: F = t

3160 bit = 316 psec 10 Mb/set

c$~ = 1.51.

The mean delay time ??d of the Ethernet can be calculated using Fig. 13.9 and Table 13.12: Td = 0.011 km. 5 psec/km + 0.005 km . 4.3 psec/km + 0.05 km - 4.8 psec/km = 0.3 psec. Table13.12 Characteristics of the optical fiber, the Cheaper-net,and the twisted pair Transfer Rate Optical fiber Cheapernet Twisted pair

Mean Length

10 Mb/set 10 Mb/set 10 Mb/s

11 m 5m 50 m

Signal Time 5 psec/km 4.3 psec/km 4.8 psec/km

In this case, Td can be neglected compared to Ft and we have: -

1

= 316psec.

peth

To obtain the service rates of the IS node (WAN or computers in a section) we use the formula: xi = ,i Pi ’ and have: (13.6)

CASE STUDIES OF QUEUEING NETWORKS

617

The values of the Xi are listed in Table 13.11. In order to get the K; we need the mean number of active computers. This value cannot be measured directly but we have the mean total number of active computers which is 170. To get an approximation, for the Ki, we divide 170 by the number of sections, which is 12 (see Table 13.4 or Fig. 13.17). This yields:

Ki For sections with

E $y

servers (multiple

= 14.17.

server node) we use:

pi= xi ml-La ’ and obtain: pi = xi

(13.7)

mipa ’

and can calculate the service rates using the utilization pZ of the servers, which was measured approximately as 90%, and the number of servers mi from Table 13.6. The values of the service rates are listed in Table 13.13. Table 13.13 Station Pa

Service rates of the computers,

servers, and the WAN

1

2

3

4

5

6

7

8

9

10

11

12

19.4

14.2

14.0

14.6

20.6

11.0

21.8

14.1

14.8

8.5

30.2

31.8

Now the closed non-product-form queueing network model of the considered communication system is completely defined. We solve it using Marie’s method (see Section 10.1.4.2). 13.1.3.4 Results The closed queueing network model from Section 13.1.3.2 together with the model parameters (as derived in Section 13.1.3.3) is solved using the queueing network package PEPSY (see Chapter 12). The PEPSY input file for this example is given in Fig. 13.14. In PEPSY, either routing probabilities pij or visit ratios ei can be used in the input file. Here we use the visit ratios, which are calculated from the originally given routing probabilities using Eq. (7.5). We have a closed queueing network, therefore we have the total number of jobs K = 170 as input. From the input file, PEPSY produces an output file, shown in Fig 13.15, with all performance measures. Since we have a non-product-form network, Marie’s method was used for the analysis. In the output file we see that the computed utilization of the servers is = 0.9, which matches with the estimated value from actual measurement, Pserv and that the utilization of the ring is p7. = 0.07, which also matches with the measured value. The mean queue length of the ring is negligible and queue

618

APPLICATIONS

fig. 13.14

PEPSY input file for the LAN example.

length at the servers is QCCZ= 3.1, GCCs= 1.9, and GCC4= 3.2. The mean response time ?? for the LAN is 0.59 sec. Now we can use this model for some experiments. For example, we can change the number of active computers K. As we can then see in Table 13.14, the server nodes are the bottleneck and the number of active computers should not exceed 250 because the queue lengths become very large. The mean response time ‘T is not influenced much by the number of active computers K in our example. Table 13.15 demonstrates the influence of changing the number of servers at server nodes. If the number of servers is reduced to m2 = 10, ms = 16, and m 4 = 9, then we have a bottleneck at the server nodes. The situation deteriorates if we further reduce the number of servers. The utilization of the ring decreases when the number of servers decreases, whereas the utilization of the server nodes increases.

CASE STUDIES OF QUEUEING NETWORKS

619

Fig. 13.15 PEPSY output file for the queueing network model of the communication system (cc; = computing center%, b, = building,).

These two experiments show that it is easy to study the influence of the variation in the parameters or the structure of the LAN once we have constructed and parameterized the queueing network model. We can use PEPSY, or any other queueing network package such as QNAP2 [VePo85] or RESQ [SaMa85], for analysis. To obtain the mean response time for a computer, its service time has to be subtracted from mean system response time T = 0.585 and we obtain for the computers in building 5, for example, the mean response time 0.54 sec.

Problem 13.6 Verify the results of the communication system model just described using another queueing network package such as RESQ or QNAP2.

620

APPLICATIONS

Table 13.14 Pcrformancc measures for the communication bers of active computers K

system for different num-

K

100

130

150

170

180

200

300

P2

0.55 0.55 0.55

0.71 0.71 0.71

0.81 0.81 0.81

0.90 0.89 0.90

0.93 0.92 0.92

0.97 0.97 0.97

0.99 0.99 0.99

0.05 0.02 0.11

0.6 0.3 0.6

1.4 0.8 1.4

3.1 1.9 3.2

4.8 2.9 4.8

7.9 8.3 9.3

50 30 43

0.56 0.043

0.56 0.055

0.57 0.063

0.59 0.070

0.60 0.072

0.64 0.075

0.94 0.077

P3 P4 g2 93

Q4 T

Pring

m2 = 14, mg = 23, rnq = 13.

Notes:

Table 13.15 Performance measures for the communication system for different number of servers m2, m3, and m4 of section 1, section 2, and section 3, respectively 7 12 7

10 16 9

14 23 13

16 18 16

18 26 17

28 46 26

0.999 0.95 0.92

0.96 0.98 0.98

0.90 0.89 0.90

0.69 0.998

0.64

0.73 0.88 0.71

0.47 0.47 0.47

G2

63 9.6 7.0

10 13 23

3.1 1.9 3.2

0.4 28 0.2

0.5 0.7 0.5

0 0 0

T

1.05 0.039

0.77 0.053

0.59 0.070

0.67 0.061

0.56 0.072

0.56 0.073

m2 m:3 m4

P2 P3 P4

Pring Note:

K = 170.

13.1.4

UNIX Kernel

We now develop a performance model of the UNIX operating system [GrBo96]. A job in the considered UNIX system is in the user context when it executes user code, in the kern context when it executes code of the operating system kernel, or in the driw context when it executes driver code to set up or perform an I/O operation. In Fig. 13.16 the life-cycle of a UNIX job is shown. The job always starts in the kern context from where it switches to the user context with probability p,,,, or to the driv context with probability pi,. After a mean service time s,,,, , the job returns from the user context to the kern context. From the driw context, where the job remains Sdrive time units, the job returns to the kern context with probability Pdrivdone or starts a I/O operation with probability 1 -Pdrivdone. After finishing the I/O, the job returns to the context driu and again remains there (with mean time Sdriv)

CASE STUDIES OF QUEUEING NETWORKS

fig. 13.16

Table 13.16

(14 w>

(a31

0 Pi0

0

621

Model of a UNIX job.

Routing probabilities

0

Pdrivdone Pdone

Puser

of the monoprocessor model

0

0 0

0 0

1 - Pdrivdone

1 0

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

and returns to the kern context with probability p&-ivdone. A job can only leave the system when it is in the kern context (with probability P&ne). The mean service time in the kern context is S&n. The I/O operation is carried out on an I/O device, whereas the remaining three activities are serviced by the CPU. 13.1.4.1 Model of the UN/X Kernel The UN IX kernel just described can be modeled by a closed non-product-form queueing network with two nodes (CPU, I/O) and three job classes, priorities, and class-switching (see Fig. 13.17). user

+O--. IO

driv

Fig. 13.17

1

The monoprocessor model.

Since we have only one processor, we call this the monoprocessor model. The three job classes correspond to the user, kern, and driv context. To obtain a closed network, we assume that when a job leaves the system a new job immediately arrives at the kern queue.

622

APPLICATIONS

A state in the routing DTMC of this model is described as a pair (node number, class number) with: node number 1: CPU, node number 240, class number 1 : h-iv, class number 2 : kern, class number 3 : user.

The routing probabilities are shown in Table 13.16. In the following we summarize the assumptions we make: l

The network is closed and there are K = 10 jobs in the system.

l

The service times are exponentially

l

The system has three job classes user, kern, driv with the priority order driv > kern > user.

l

Class switching is allowed.

l

Jobs in the context user can always be preempted by driv jobs. Other preemptions are not possible.

l

The time unit used in this section is msec.

l

The following parameter values are used:

l

and kern

= varied from 0.25 to 20.0, Skern= 1.0,

pi0 = 0.05, Pd one -- O.OIL/O.OO5,

Suser

Pdrivdone

Sdriv

= 0.4,

distributed.

= O-5.

The I/O system consists of several devices that are assumed to form a composite node (see Section 8.4) with load-dependent service times (which were measured from the real system) : sio(l) = 28.00, sio(2) = 18.667, Sio(3) = 15.555, Sio(4) = 14.000, sio(5) = 13.067,

sio(6) Sio(7)

sio(8) Sio(9) Sio(l0)

= = = = =

12.444, 12.000, 11.667, 11.407, 11.200.

The parameters Suser, Skern, and S&iv are the mean service times of class i = 1,2,3) at the CPU (node l), and sio( k) is the mean load-dependent service time of class 1 jobs at the I/O devices (node 2). At the CPU (node 1) we have a mixture of preemptive and non-preemptive service strategy, while at the I/O devices (Node 2) we have only one job class with the FCFS service discipline. The following concepts are used to solve this model: (i

CASE STUDIES OF QUEUEING NETWORKS

Fig. 13.18

623

The transformed monoprocessor model.

l

Chain concept (see Section 7.3.6)

l

Extended shadow technique with mixed priority strategy and class switching (see Section 10.3.2.2).

Since we have three job classes, we split the CPU into three CPUs, one for each job class (see Fig. 13.18). Approximate iterative solution of the resulting model can be carried out by PEPSY or SHARPE.

0

CPU

Fig. 13.19

The master slave model.

13.1.4.1.1 The Master Slave Mode/ For this model, shown in Fig. 13.19, we have the same basic assumptions as for the monoprocessor model but now we also have two APUs. An APU is an additional processor (associate processing unit) that works with the main CPU. It is assumed that only jobs in the user context can be processed at the APU. Since this is the only job class that can be processed at the APU, no preemption can take place there. If the APU as well as the CPU are free and there are jobs in the user queue, then the processor (CPU, APU) on which the job is served is decided randomly. If we apply the shadow transformation to the original model, we get the transformed master slave model, shown in Fig. 13.20.

624

APPLICATIONS

I

APU

0(I

user

6

CPUuser CPUkern CPUdriv

& 0

Fig. 13.20

The transformed master slave model.

Fig. 13.21

The associated processor model.

13.1.4.1.2 The Associated Processor Model The assumptions for the associated processor model, shown in Fig. 13.21, are the same as for the master slave model but now jobs in the kern context can also be processed on the APU. Since jobs in kern context have higher priority than jobs in user context, preemption can also take place at the APU. The model needs to be transformed so that it can be solved approximately using the shadow technique in combination with the MVA. The transformed associated processor model is shown in Fig. 13.22. We now solve the three non-product-form network models 13.1.4.2 Analysis using approximate techniques and compare the results to the ones obtained from the (exact) numerical solution of the underlying CTMC. We show the computation of the performance measures for the monoprocessor model step

CASE STUDIES OF QUEUE/NC NETWORKS

0

625

CPUuser CPUkern

CPU&-iv

Fig. 13.22

The transformed associated processor model

by step. The procedure is analogous only the results are given. 13.1.4.2.1 The Monoprocessor Model tains three job classes and two nodes: class 1 : &iv, class 2 : kern, class 3 : user,

for the other two models and therefore

The model,

given in Fig. 13.17, con-

node 1 : CPU, node 2 : I/O.

order of the job classes is driu > kern > user. Jobs in the context by driw and kern jobs. Other preemptions are not possible. The chosen numerical parameter values are: The priority

user can always be preempted P11,12

= Pdrivdone

P12,ll

=

P12,13

= Puscr

P21,ll

=

Sll

Pi0 = 0.05, --

0.94,

Pll,21

=

1 - Pdrivdone

P12,12

= Pdone

P13,12

=

1.0,

1.0,

= S&-iv

5’12=

= 0.4,

Skerp

=

0.5,

E = 0.001,

=

1.0,

N = 5 jobs,

s13

= Sllscr = 1.5,

~21

= sio = load dep.

= 0.01,

= 0.6,

626

APPLICATIONS

Now the analysis nine steps:

of the monoprocessor

Compute

the visit ratios

ei, using Eq. (7.15):

ell = edriv = 12.5, = euscr = 94,

e13

Fig. 13.23

model can be done in the following

e12

=

ekern

= 100,

e21 = e;, = 7.5.

The routing DTMC for the monoprocessor model.

Determine the chains in the network. The DTMC state diagram for the routing matrix in the monoprocessor model is given in Fig. 13.23. As we can see, every state can be reached from every other state, which means that we have only one chain in the system. Transform the network into the shadow network (see Fig. 13.18). The corresponding parameter values for the transformed model are: 4511= 0.5,

s22 = 1.0,

s33

ell = 12.5,

e22 = 100,

e33 = 94,

Compute

sA1 = load dep.

= 1.0,

e41 = 7.5.

the visit ratios ezq per chain and recall that we have only

one chain:

Then we get: eTl

=

ell

ell

e33

ezl = G

Compute

=

e22

1, eG1

= 7.52,

the scale factors

=

z e41

eil = G

8, =

= 0.6.

air, using the equation:

CASE STUDIES OF QUEUEING NETWORKS

and get: a11

ell

=

a12

= 1.0,

ell

+ el2 el2

+ e13

ell

+ el2 el3

+ ~13

=

Q13 = a21

ell +

el2 e21

+

e13

e21 + e22 + e23 e22 a22

= e21 + e22 + e23 e23

a23

= e21 + e22 + e23

1-1

= 0,

Cl!32 =

XZ 0,

a33

=

e31 + e32 + e33 e33

Q41 = e41 + e42 + e43 e42

1.0,

a42

= e41 + e42 + e43 e43

= 0,

a43

=

1 0, = 0,

= 1.0, = 1.0, = 0, =

0.

e41 + e42 + e43

Compute th e mean service times for the chain: srq zz -

1

=

t%q

ST1= 0.5, m] Iteration

=

e31 + e32 + e33 e32

e31 + e32 + e33 e41

= 0,

=

e31

Q31 =

sll = 1.0,

c

Si?” ’ air,

rE7rp

s& = 1.5,

s& = load dep.

Start the shadow iterations: 1: pi1 = 0.037,

p;1 = 0.594,

si;‘” = 0.5,

p& = 0.838,

pi1 = 0.659,

Ss;;‘” = 1.03842,

x;, = 0.075,

x;, = 0.594,

sS$;‘*= 4.06504,

x;, = 0.558,

xl;, = 0.045,

-(‘)* S41

pT1 = 0.016,

p& = 0.27,

p& = 0.994,

pi1 = 0.289,

sg* = 0.5, -(2)* = 1.01626, S21

XT1 = 0.033,

A;, = 0.26,

SE’* = 2.10084,

A;, = 0.245,

xi, = 0.02,

-(‘)* S41

pi1 = 0.030,

,o& = 0.481,

siy* = 0.5,

pG1= 0.935,

pi1 = 0.525,

g*

= 1.03092,

XT1 = 0.033,

x;, = 0.473,

-(3)* S31

= 3.16373,

Xi1 = 0.245,

xi, = 0.02,

-(3)* S41

= load dep.

= load dep.

Iteration 2:

= load dep.

Iteration 3:

627

628

APPLICATIONS

Iteration 15: Finally, after 15 iterations, the solution converges and we have the following results. Sll ,.(15)*

--

pi1 = 0.435,

@*

= 1.0256,

x;, = 0.392, xl;1 = 0.029,

S31 -(15)*

= 2.61780,

-(15)* S41

= load dep.

pi1 = 0.025,

&

/I& = 0.969, X;cl = 0.049, A;, = 0.368,

= 0.402,

0.5,

Retransform the chain values into class values:

x11 = allA;, = 0.049, x22 = cQ&,

= 0.392,

X33 = CX~~X;,

= 0.368,

x41

=

= a4&

0.029.

If we rewrite the results of the shadow network in terms of the original network, we get the following final results:

~~;ppro”) = 0.049, x~;pp’“) = 0.392, J@prox)

The overall throughput

= 0.368,

xFprox)

= 0.029.

of the network is then given by:

x(wp rox) = 1000 ’ pdone * ~$prox’

= 3.92.

Verify the results: To verify the results, we construct and solve the underlying CTMC using MOSES. The exact overall throughput is given by: X(exact) = 3.98. The difference between the exact and the approximate throughput

is small.

The other two models were also analyzed using the extended shadow technique with class switching in a similar manner. The overall system throughput of the UNIX models is plotted as a function of the user service time in Fig. 13.24. There is nearly no difference between the approximate values and the exact values obtained from the numerical solution of the CTMC for the chosen parameter set. As expected, the associated processor model has the highest system throughput. In a more detailed study it was found that for the master slave model the throughput cannot be increased any more by adding additional APUs because the CPU is the bottleneck. For systems that spend a lot of time doing system jobs (UNIX is said to spend approximately 50% of its time doing system work) it is worth having a parallel system kernel. For more I/O-intensive jobs, the number of

CASE STUDIES OF QUElJElNG NETWORKS

Fig. 13.24

Throughput

629

for the different models covered in Section 13.1.4.

peripheral devices must also be increased, otherwise the processors have to wait too long for I/O jobs to be completed. In this case it is not worth having a multiprocessor system. Another very important result from this study is that for suSer> 3 msec, the master slave and associated processor model behave nearly the same and only for s,,,, < 3 msec does the associated processor model gives better results than the master slave model.’ However, it does not seem to matter as to which of the two (an associated processor configuration or a master slave one) configurations is used. On the other hand, if the system is used in a real-time environment where response times are required to be extremely short, the associated processor model delivers better performance than the master slave configuration and it is worthwhile to have a completely parallel kernel. The computation time to solve the model was approximately 45 minutes if the CTMC method is used. Using the shadow technique, the results were generated within seconds for the entire parameter set. By contrast, when this non-product-form network was solved using DES, it took nearly 120 hours of computing time. The impact of priorities is shown for the master slave model in Table 13.17. Here pi, is varied while p&ne is constant. On the right-hand side of the table the throughput for the network without priorities (nonpre), the original values with priorities (orig = obtained from numerical solution of the CTMC), and approximate values (approx = obtained using the shadow

‘This result means that if the system worthwhile to have a parallel kernel.

will

be used

for very

time

consuming

jobs,

it is

630

APPLICATIONS

approximation) are given. As can be seen in the table, for some parameter values the impact of priorities cannot be neglected. Tab/e13.17 Impact of the Suer

Flexible

0.9 0.01 0.001 0.9 0.01 0.001 0.9 0.01 0.001 0.9 0.01 0.001 0.9 0.01 0.001

Production

@done = 0.01) Throughput

Pi0

0.50 0.50 0.50 1.00 1.00 1.00 1.50 1.50 1.50 2.00 2.00 2.00 2.50 2.50 2.50

13.1.5

priorities

Nonpre

Approx

Orig

0.6593 9.0049 9.0897 0.6593 8.2745 8.3398 0.6593 7.6517 7.7027 0.6593 7.1078 7.1489 0.6592 6.6170 6.6485

0.6594 9.8765 9.9875 0.6594 9.8736 9.9859 0.6594 9.8194 9.9395 0.6594 9.5178 9.6326 0.6593 8.8123 8.8802

0.6593 9.8765 9.9875 0.6593 9.8716 9.9832 0.6592 9.7666 9.8983 0.6592 9.4700 9.4000 0.6592 8.6144 8.5100

Systems

Queueing network models combined with numerical solution methods are also a very powerful paradigm for the performance evaluation of production systems. In this section we demonstrate how to model and analyze such systems using open and closed queueing networks. 13.1.5.1 An Open Network Model Consider a simple production system that can be modeled as an open queueing network. The system contains the following stations (Fig. 13.25): l

A load station where the workpieces are mounted on to the pallet (LO).

l

Two identical lathes (LA).

l

Three identical milling machines (M).

l

A transfer system (T) that does the transfer between the stations and consists of two automatically controlled vehicles.

l

A station to unload the pallet (U), which removes the workpieces from the production system.

The identical milling machines and lathes are modeled by multiple server nodes. With these assumptions we obtain the queueing network model shown

631

CASE STUDIES OF QUEUEING NETWORKS

Fig. 13.25

Layout of a simple production

system.

Source

Fig. 13.26

Open network model of the production

system shown in Fig. 13.25.

in Fig. 13.26. The production system produces different products and therefore the machines are used in several different ways. For example, 60% of the workpieces that leave the milling machine are passed to the station that unloads the pallet, the rest are transferred to the lathes. Table 13.18 contains the probabilities qij, i = LO, M, LA, j = M, LA, U that the transfer system T moves workpieces from a station i to station j. To obtain from the transfer probabilities qij the probabilities of routing from the transfer system T to the lathes PT,LA, milling machines PT,M, or the unload station pT,U, we have to weight the individual probabilities qij with the individual arrival rates A;, i = LO, M, LA. The arrival rate XT to the transfer system can be easily obtained from Fig. 13.26:

AT

=

ALO

+

h4

+

ALA,

(13.8)

632

APPLICATIONS

Table 13.18 Routing table for the transfer system of the production system '

\i LO M LA

MLA

U

0.5 0 0.7

0.5

0

0.4

0.6

0

0.3

and it follows that: PT,M

ALO

=

- XT

ALO

PT,LA = Xr

ALA

PT,U = -

XT

ALA * qLO,M $- x T

’ qLA,M,

AM + x'qM,LA, T

'qLO,LA * qLA,U

AM XT

i-

(13.9)

* qMJJ.

Then, using the values from Table 13.18:

PT,M= & (ALO* 0.5 -k ALA * 0.7)) PT,LA = $(A PTJJ

=

&

LO ’ 0.5 + AM - 0.4) , (ALA

*

0.3 + &f * 0.6) ,

we finally obtain the matrix of routing probabilities pij for the queueing network model (see Fig. 13.26 and Table 13.19). In order to apply Jackson’s Table 13.19 Routing matrix for the queueing model of the production system

\i

j

outside

LO LA M M u T

outside

0 0 0 0 0 1 0

theorem for open product-form following assumptions:

LO

LA

1 0 0 0 0 0 0

0 0 0 0 0 0 ?‘LA

M

UT

000 001 001 001 001 000 ?‘M

PU

0

networks (see Section 7.3.4), we make the

CASE STUDIES OF QUEUEING NETWORKS

a The service times at the stations and the interarrival station LO are exponentially distributed. l

The service discipline at each station is FCFS.

l

The system is stable.

633

times at the load

Table 13.20 Arrival rates Xoi, service rates pi (in l/h), and number of servers m, for the model of Fig. 13.26 i

XOi

Pi

LO LA M u T

15 0 0 0 0

20 10 7 20 24

mi

1 2 3 1 2

The service rates pi, the arrival rates Xei, and number of servers rni are listed in Table 13.20. Now we use Eq. (7.1) and the values of the routing probabilities from Table 13.19 to determine the arrival rates Xi: Ai = XOi + C

Xj

i, j = LO, LA, ikl, U, T,

* pjiy

(13.10)

and obtain: XL0 = XOLO = 15, ALA = XT - PT,LA = XLO - 0.5 + AM - 0.4 = 14.58, AM = XT ‘PT,M = ALO * 0.5 •k ALA - 0.7 = 17.71, Au = XT * PTJJ = ALO = 15, XT = ALO + ALA + AM = 47.29. Performance measures, which were calculated using Jackson’s theorem, are listed in Table 13.21. We see that the transfer system T is heavily loaded and has a very long queue, and that its utilization is nearly 100%. If we use an additional vehicle, the utilization and the queue length are reduced substantially to PT = 0.66,

QT

=

0.82,

WT = 0.02.

The work in progress (WIP, the mean number of workpieces in the system) and the mean response time ?II for both cases are listed in Table 13.22. To improve the performance we could increase the number of milling machines because they have a very high utilization, PM = 0.84, or the number of lathes. The results for these changes are shown in Table 13.23. A further increase in

APPLICATIONS

634

Table 13.21 Performance measures for the production system

LO LA A4 u T

2.25 1.66 3.86 2.25 65.02

0.15 0.11 0.22 0.15 1.38

0.75 0.73 0.84 0.75 0.985

Tab/e 13.22 Work in progress (WIP) and mean response time T for 2 and 3 vehicles in the transfer sys km mT

2 3

WIP

T-

82.5 18.3

5.5 1.2

the number of the vehicles in the transfer system has a negligible influence on the performance. Maximum improvement can be achieved by increasing the number of milling machines, because that station is the bottleneck (p; = 0.84, see Table 13.21). Although we have used a very simple model to approximate the behavior of the production system, it does provide insight into the system behavior and enables us to identify bottlenecks. 13.1.5.2 A Closed Network Model Now we consider the example of a production system that can be modeled as a closed multiclass queueing network [SuHi84]. The production system consists of (see Fig. 13.27): a Two load/unload stations (LU). l

Two lathes with identical tools (LA).

Tab/e 13.23 WIP and T for different numbers of vehicles, milling machines, and lathes mT

mLA

mM

3 4 3 3 3

2 2 3 2 3

3 3 3 4 4

WIP

T

18.3 17.6 16.9 15.0 13.6

1.22 1.18 1.12 1.00 0.90

CASE STUDIES OF QUEUEING NETWORKS

l

Three machine

l

A central

Fig. 13.27

centers with different

transfer

tools (M).

system with eight transporters

Layout of the production

635

(T).

system for a closed multiclass network model.

In the production system, workpieces for three different product classes are manufactured. A fixed number Ki of pallets for each class circulates through the system. In the first class the workpieces are mounted on the pallets in the LU station then they are moved by the transfer system either to the lathes LA or to the third Ms. Finally, they are shipped back to the LU station where they are unmounted. For workpieces of the other two classes, the procedure is similar, but they use machine MJ and MS, respectively. The load/unload station LU and the lathes LA can be modeled as a multiple server node and the machines Ma, Md, and n/r, as single server nodes. Since there are more transporters in the transfer system T than circulating pallets, it is modeled by an IS node. The parameters for the model are summarized in Table 13.24.

Table 13.24 Station, LU LA M3 M4

M5 T

The numbers are given by:

Parameter values for the stations mz

l/PLz

2 2 1 1 1 8

7.5 28 12 30 15 2.5

el,

e2%

2 0.5 0.5 0 0 2

2 0 0 1 0 2

of pallets Ki, which are the numbers

K1 =4,

K2 = 1,

2 0 0 0 1 2

of workpieces

in class i,

K3 =2.

With these parameters we obtain the queueing network models for these three classes, given in Fig. 13.28. Furthermore we assume that each network fulfills the product-form assumptions. Now we have all the necessary input param-

636

APPLICATIONS Class 1:

Class 2:

Class 3:

=rl’lq-@ ’ outside

Fig. 13.28

0.5

Closed queueing model of the production

system shown in Fig. 13.27.

eters and can use any solution method for closed product-form queueing networks such as MVA, convolution, or SCAT to obtain performance measures. In Table 13.25, the throughput and the mean system response time for the different classes are shown. The total throughput is 0.122 workpieces/min or 7.32 Table 13.25 Class 1 2 3

Performance measures for workpieces Throughput 0.069 0.016 0.037

Mean Response Time 58.35 63.88 53.77

workpieces/h. For other interesting performance measures of the machines, see Table 13.26. Using the queueing network model, the system can be optimized. In Table 13.26 we can see that the load/unload stations are the bottleneck and, therefore, we increase the number of these stations to three and four, respectively. The results for the modified system models are given in Table 13.27,

CASE STUDIES OF QUEUEING NETWORKS

Tab/e 13.26

Station,

Performance measures for the stations

m,

LU LA M3 M5

T

Mean number

Mean Queue

in Service

Length oi

1.82 0.96 0.41 0.47 0.56 0.61

1.68 0.12 0.19 0.00 0.16 0.00

pi

2 2 1 1 1 8

M4

637

0.91 0.48 0.41 0.47 0.56 0.08

Tab/e 13.27 Utilizations of the stations for different number of servers at the LU stations and routing probabilities of Table 13.28

LU LA M3 M4

M5 T Pmax - Pmin

1

2

3

4

4a

4b

4c

4d

0.997 0.27 0.23 0.26 0.30 0.04 0.77

0.91 0.48 0.41 0.47 0.56 0.08 0.50

0.73 0.58 0.49 0.57 0.68 0.092 0.240

0.58 0.61 0.52 0.59 0.72 0.096 0.200

0.55 0.60 0.56 0.68 0.52 0.09 0.16

0.56 0.60 0.54 0.64 0.62 0.09 0.10

0.56 0.60 0.56 0.59 0.62 0.08 0.06

0.77 0.82 0.77 0.82 0.85 0.129 0.08

columns 2, 3 and 4. In column 1, the reduced to one. In this case the system at LU, the mean response time is greater length at LU is 3.23. For rnLu = 3, the

Tab/e 13.28

I’23 E ii

0.1 0.2 0.2

number of load/unload stations is works, but with a severe bottleneck than 100 units, and the mean queue utilization of the LU station is still

Routing probabilities

I’24

P34

0.9 0.8 0.1 0.2 Kl = 0.8 8, K2 = 2, 0.1 K3 = 4

P35

0.9 0.8 0.9

,OLU = 0.73, and th e system is still unbalanced, which means that the utilization of the different stations differ considerably, as seen in the last column of Table 13.27. For rnLu = 4, the system is more balanced, but now PM5 = 0.72 is too high. To get a more balanced system, a fraction of the workpieces of class 2 are serviced at station A& (see Fig. 13.29). Now the system is better balanced especially in case 4c, but the utilizations are relatively low (pi E 0.6).

638

APPLICATIONS

outside Class 3:

$lgy@&g outside

Fig. 13.29 with different

Queueing routing

.

network model of the production for class 2 and 3.

system shown in Fig. 13.28,

Therefore, we double the number of pallets in the system (K1 = 8, E(2 = 2, and Ka = 4) and obtain the values of the utilization in column 4d. The system is now balanced at a higher level of utilization (pi z 0.8). Increasing the number of LU servers leads to a higher throughput and a lower mean response time (see Table 13.29). If we increase the number of pallets to Kr = 8, K2 = 2 and KS = 4, the mean response time increases along with the throughput (see Table 13.30). Performance

measures with four LU servers

Class

Throughput

Mean Response Time

1

0.086 0.020 0.048

46.29 50.61 41.97

Table 13.29

2 3

Problem 13.7 Verify the results of the closed queueing network model of the production system by hand computation and by using the SHARPE software package.

13.2

CASE STUDIES OF MARKOV

CHAINS

In this section we show the use of CTMC models either constructed directly or via a higher-level specification.

CASE STUDIES OF MARKOV CHAINS

639

Table 13.30 Performance measures with four LU servers and a higher number of pallets Class

Mean Response Time

Throughput

1 2 3

0.117 0.026 0.063

68.21 76.56 63.54

Notes: K1 = 8, K:! = 2, KS = 4.

13.2.1

Wafer

Production

System

In this section we formulate and solve a CTMC model of the photolithographic process of a wafer production system. The system is specified using a queueing network formalism, although for this product-form network we use the package MOSEL to automatically generate and solve the underlying CTMC. In [Rand861 the photolithographic process is divided into the following five steps: 1. Clean the wafer. 2. Put on photoresist. 3. Bake wafer and remove solvent by evaporation. 4. Align mask, and illuminate

and develop the wafer.

5. Etch the wafer and remove photoresist. The photolithographic

processing is carried out by three machine types:

l

The spinning machine does the first three steps (machine type 1).

l

The masking machine does step four (machine type 2).

l

The etching

machine

does step five (machine type 3).

From the first machine type, three machines are available. From the second machine type, we have machines from different producers. These machines are called masking 1 and masking 2. The third machine type does the steps of etching and removing the photoresist. The job of removing the photoresist is done by machine five in case the wafer has not been produced correctly. Since three layers of photoresist have to be put on, the wafer needs to be processed again after a successful masking and spinning. In Fig. 13.30, the open queueing network model of the wafer production system is shown and in Table 13.31 its parameter values are given.

640

APPLICATIONS

Etching,

Remove

Good Wafer Remove Photoresist

Fig. 13.30 13.2.1.

Open network model of the wafer production system discussed in Sec-

tion

In the representation of the queueing network model we use node 6 as a dummy node with zero service time. We use node 6 in order to easily determine the routing probabilities, but it is, of course, also possible to determine the routing probabilities without the introduction of node 6. Since it is an open network, we limit the number of wafers in the system to K so that the underlying CTMC is not infinite. The MOSEL specification of the network is shown in Fig. 13.31; note the use of the loop construct to reduce the size of the specification. WIP is the work in progress, i.e., the mean number of wafers in the system. This value needs to be as small as possible for good system performance. Results of the analysis are shown in Fig. 13.32. Here we plot the mean response time as a function of the number of machines at station 1. There are several things that we can learn from the plot: l

Having more than five machines at station 1 does not reduce the mean response time very much. Therefore it is not cost effective to have more than five machines at station 1.

l

Increasing the number of machines at stations 2 and 3 from two to three reduces the mean response time considerably, while adding more machines is not any more cost effective.

l

We get a large decrease in the mean response time if we increase the number of machines at station 1 from three to four and if we increase the number of machines at stations 2 and 3 from two to three.

CASE STUDIES OF MARKOV CHAINS

641

Tab/e 13.31 Parameter values for the wafer production system discussed in Section 13.2.1

ml m2 mg mq

= 2...4 = 2...4 = 2

m5

=1

p12 = 0.5 p13 = 0.5 p24 = 0.9

=3...10

p25

= 0.1

p34

=

0.9

= 0.1 pqg = 0.1 p35

p4t3 =

0.9

PGO = 213 PGl

It is possible to extend

13.2.2

l/3

this model to allow for machine

Problem reward

=

13.8 Formulate the wafer production net and solve it using SPN P. Polling

failures

problem

and repairs. as a stochastic

Systems

Polling systems are very important in modeling computer and communication systems. The generic polling model is a system of multiple queues visited by a single server in cyclic order (see Fig. 13.33). For an exposition and performance models of polling systems, we refer the reader to [TakaSO]. In this subsection we wish to illustrate the use of SPNs in automatically generating and solving the underlying CTMC for polling system performance models. Our treatment is based on the paper by Ibe and Trivedi [IbT’rSO]. In Fig. 13.34, the GSPN for a three-station finite population single service polling system is shown. The potential customers that can arrive at station j are indicated by tokens in place Pjr, j = 1,2,3. This place contains Mj tokens where iVj denotes the population size of customers that potentially arrive at the station. The firing rate of the transitions tlj is marking-dependent, meaning that when the number of tokens in place Pjr is Ic, then the firing rate of tlj is given by Xjlc, 0 < Ic < LV,?~.A marking-dependent firing rate is represented by the # symbol and placed next to transition tlj. A token in place E’jp represents the condition that the server is polling station j; a token in place PUB represents the condition that a customer has arrived at station j, and Pjs represents the condition that the server has arrived at station j. The server commences polling station j + 1 after finishing service at station j or found no customer there when it arrived. Whenever the timed transition tlj fires, a customer arrives at station j. The server has finished polling station j when the timed transition t,j fires. Such a GSPN can be easily input to any of the SPN tools such as SHARPE or SPNP, whereby the underlying CTMC will be automatically generated and solved for steady-state or transient

642

APPLICATIONS

Fig. 13.31

MOSEL specification of the wafer production

system.

behavior, and resulting performance measures calculated. We refer the reader to [STPSB] and [IbTrSO]. Other kinds of polling schemes have also been considered by [IbTrSO]. In the gated service polling system, only those customers are served by the server that arrive prior to the server’s arrival at station j. The SRN model of station 1 in a finite population gated service polling system is shown in Fig. 13.35. The difference between the GSPN model of the single service polling system is that now a variable multiplicity arc (indicated by the zigzag sign) is needed. Therefore, the new model is a stochastic reward net model that cannot be solved by pure GSPN solvers such as SHARPE or GreatSPN [STP96, ABC+95]. However, SPN P is capable of handling such models. For further details we refer the reader to [IbT!rSO] and [ChTr92].

CASE STUDIES OF MARKOV CHAINS

* Station

2 and

3 each

have

two

p

Station

2 and

3 each

have

three

I)-----o

Station

2 and

3 each

have

four

Number

Fig. 13.32

of machines

at Station

643

machines machiucs machines

1

Results for different number of machines.

The SRN (or GSPN) models can be solved using SPNP. The SPNP input file contains the specification of timed transitions and corresponding firing rates, immediate transitions, the input and output arc for each transition, and the initial number of tokens in each place. From this information SPNP generates the reachability graph, eliminates all vanishing markings, and constructs the underlying CTMC. In order to obtain the steady-state probability of each marking, a combination of SOR and the Gauss-Seidel method is used. It is possible to obtain the mean number of tokens in each place as well as more general reward-based measures. Furthermore, transient and sensitivity analysis can also be carried out.

.. I1o3 Fig. 13.33

A polling queue.

644

APPLICATIONS

I

L

I P2P

Y--

Fig. 13.34

GSPN Model

for the single service polling

system.

CASE STUDIES OF MARKOV CHAINS

I Q

Fig. 13.35

P2P

72

from

645

t,l-r .1

SRN Model of gated service polling system.

Problem 13.9 Specify and solve the GSPN model of the single server polling system using SHARPE. Determine the size of the underlying CTMC as a function of the number of customers &,!I = iU2 = MS = M. Determine the steady-state and the transient performance measures. Problem 13.10 Specify and solve the SRN model of the gated service polling system using SPNP. Determine the steady-state and transient performance measures. Problem 13.11 Extend the polling model of Fig. 13.34 to an Erlang-3 distributed polling time. Run the resulting model on both SHARPE and SPNP. 13.2.3

Client Server Systems

Consider a distributed computing system with one file server and N workstations, interconnected by a LAN. Because the communication medium is shared by several workstations, the server and each workstation should have access to the network before it starts transmitting request/reply messages. It is assumed that a workstation does not start generating a new request until it has received the reply of its previous request. This assumption is based on the fact that in many situations future operations at a client workstation

646

APPLICATIONS

depend on the outcome of the current operation (current service request). As long as a workstation’s request is outstanding, it can continue its local processing. Our model deals only with the aspect of a user’s processing that requires access to the server. Our exposition here is based on [ICT93]. In our system model, we assume that each time the server captures the access right to the network it transmits at most one reply message. The order of service at the server is FCFS. It is important to distinguish between the client server models here and the single buffer polling system presented in Section 13.2.2. The difference is that in a single buffer polling system the station does not wait for reply to its message. Thus in a polling system, as soon as a station transmits its message, it is ready to generate a new one after the reply to the last transmitted message has been received. The interdependencies of the access control of the network and workload between the client workstations and the server also make the analysis of client server systems different from the traditional analysis of plain LANs. We consider client server systems based on a token ring network. For another model with CSMA/CD, see [ICT93]. The first step of the analysis is to find an appropriate SRN model for the system. Since there might be a confusion between the word token in a Petri net or in a token ring network, we refer to token in a ring network as network token and to the token associated with the SRN as the PN token. We assume that the time until a client generates a request is exponentially distributed with mean l/X. The time for the network to move from one station to the next station (polling time) is also assumed to be exponentially distributed with mean l/y. The processing time of a request at a server is exponentially distributed with mean l/p, and the time required for the server to process a request is assumed to be exponentially distributed with mean l/p. The system consists of N clients and one server. We note that the assumption of exponential distribution can be realized by resorting to phase-type expansions. As can be seen in Fig. 13.36, we consider a tagged client and lump the remaining clients into one super-client. This superclient subsystem enables us to closely approximate (N - 1) multiples of the single client subsystem so that we reduce the number of states of the underlying CTMC. The part of the SRN model that deals with the tagged client is described in Fig. 13.37. The place PTI contains one PN token and represents the condition that the client is idle (in the sense the client does not generate a network-based request). The firing rate of the timed transition tt, is X. Thus, the firing of tt, represents the event that the client has generated a request. The place PTA represents the condition that the network token has arrived. Suppose that there is a PN token in PTS. Then if there is a PN token in PTA, the timed transition tt, whose mean time to fire is the mean request transmission time is enabled; otherwise the immediate transition sr fires. The place PSI represents the condition that the client’s request has arrived at the server, and the place PTW represents the condition that the client is waiting to receive a reply from the server. The place P sp represents the condition that the client

CASE STUDIES OF MARKOV CHAINS Server

fig. 13.36

A possible configuration

647

Tagged Client

of network

relative

to the tagged client.

has finished transmitting its request, if it is reached through tts, or the client has no request to transmit, if it is reached via si. The timed transition t,, represents the event that the network token is being passed to its neighbor, the server in this configuration. The place PSS represents the condition that the server has received the network token and can commence transmitting a reply, if one is ready. It should be noted that in this model, a network token is released from a station as soon as it finishes transmitting its packet. This operation of the token ring network is called the single-token operation and is used in IEEE 802.5 token ring networks [Ibe87]. Single-token operation requires a station that has completed its packet transmission to release a network token only after its packet header has returned to the station. In the case that the round-trip delay of the ring is lower than the transmission time, the two schemes become identical. Let us consider now the superclient subsystem. The corresponding SRN for this subsystem is shown in Fig. 13.38 and can be interpreted as follows: Place Par initially contains N - 1 PN tokens and represents the condition that no member of the superclient subsystem has generated a request. The number of tokens can generally be interpreted as the number of idle members of the subsystem, The firing rate of the timed transition t,, is markingdependent so that if there are 1 PN tokens in place POT, the firing rate of t,, is given by 1 . X(0 5 1 2 N - 1). The places POW, POS, PTP, and POA play the same role respectively as PTW, PTS, Psp and PTA of Fig. 13.37. Similarly, the transitions ~2, ttp, and t,, correspond respectively to si, tsp, and tt, of Fig. 13.37. Furthermore, the two additional places Pc2 and Pc1 are introduced and used to keep a count of the number of times the PN

648

APPLICATIONS

L k---l 0 SP

Fig. 13.37

0

0

pss

SRN for tagged client subsystem.

PTS

Fig. 13.38

SRN for the superclient subsystem in token ring network.

CASE STUDIES OF MARKOV CHAINS

649

token has gone through POS. As soon as each member of the superclient subsystem has been polled, a PN token is deposited in both places Pcl and Pc~. Note that P ~2 receives a PN token only after a walk time has been completed while Pcl receives the PN token immediately after a request, if one is transmitted, or prior to the commencement of the polling of the next member, if no transmission takes place. A regular arc of multiplicity N - 1 from place Pc1 is input to transition s4 and an inhibitor arc of multiplicity N - 1 is input from Pc~ to the immediate transition ss. Transition ss is enabled if a PN token is in place Pc2 and fewer than N - 1 PN tokens are in place Pci. This represents the condition that the network token leaves the superclient subsystem if fewer than N - 1 clients have been polled. If the number of PN tokens in place Pc~ reaches N - 1, implying all members of the subsystem have been polled, then with a PN token in Pc~, s4 fires immediately and a PN token is deposited in PTS. In this situation the tagged client now has access to the network after all members of the superclient subsystem had their chance to transmit their requests. ’

0

PTI

.

0 N-l

POI Kg. 13.39 SRN for the server subsystem.

The SRN for the subsystem is shown in Fig. 13.39. The places Pop, PSS, and PSA serve the same purpose as Psp, PTS, and PTA in Fig. 13.37. A token in place PSI represents the condition that the server has received a request, while the place PTw (POW) represents the condition that a client

650

APPLICATIONS

from the superclient subsystem (the tagged client) is waiting for a reply to its request. The firing rates of the timed transitions top, t,,, and t,, are given by y, p, and X respectively. The condition that the server has completed serving one request is represented by the place PSW and a determination is to be made regarding whose request was just serviced. In order to separate the requests from the superclient subsystem the server receives before that of the tagged client from those the server receives after that of the tagged client, the immediate transition s6 is used. To denote that the input arc from POW to .!96has a variable multiplicity, the zigzag sign on that arc is used. Let us assume POWcontains Ic PN tokens, 0 < Ic 5 N - 1, and PTW has no token. Then the immediate transition s6 is enabled and fires immediately by removing all Ic PN tokens from POWand deposits them in POH. The number of clients from the superclient subsystem whose requests were received by the server before that of the tagged client is therefore given by the marking of POH. The tagged client’s request cannot be replied to until requests from the superclient subsystem that arrived before it have been replied to, or stated differently. Due to the FIFO queue at the server, the immediate transition ss cannot fire as long as there is a PN token in POH. As soon as the immediate transition s7 fires, a PN token is deposited in POI, which means that a waiting member of the superclient subsystem becomes idle again. The complete SRN model for the system is shown in Fig. 13.40. Problem 13.12 Specify and run the client server SRN using the SPN P package. Compute the average response time as a function of the number of stations N. In each case, keep track of the number of states and the number of non-zero transitions in the underlying CTMC. Problem 13.13 Extend the previously described model to allow for an Erlang-2 distributed polling time. Solve the new SRN model using SPNP. 13.2.4

ISDN

Channel

13.2.4.1 The Baseline Mode/ In this section we present a CTMC modeling a simplified version of an ISDN channel. Typically, heterogeneous data are transported across an ISDN channel. In particular, discrete and continuous media are served at the same time, although different qualities of service requirements are imposed by different types of media on the transport system. It is well known that voice data is very sensitive to delay and delay variation (jitter) effects. Therefore, guarantees must be given for a maximum endto-end delay bound as far as voice data is concerned. On the other hand, continuous media streams such as voice can tolerate some fraction of data loss without suffering from perceivable quality degradation. Discrete data, in contrast, are very loss sensitive but require only weak constraints as far as delay is concerned. Trade-off analysis is therefore necessary to make effective use of limited available channel bandwidth.

CASE STUDIES OF MARKOV CHAINS

pTP

Fig. 13.40

SRN for the client server system (N > 1).

651

652

APPLICATIONS

In our simplified scenario, a single buffer scheme is assumed that is shared between a voice stream and discrete data units. For the sake of simplicity, the delay budget available for voice packets to be spent at the channel is assumed to be just enough for a voice data unit transmission. Any queueing delay would lead to a violation of the given delay guarantee. Therefore, voice packets arriving at a non-empty system are simply rejected and discarded. Rejecting voice data, on the other hand, is expected to have a positive impact on the loss rate of data packets due to sharing of limited buffer resources. Of course, the question arises to what extent the voice data loss must be tolerated under the given scenario.

Fig. 13.41

CTMC

modeling

an ISDN channel with voice and discrete data.

Table 13.32 Parameter values for the ISDN channel in Fig. 13.41 Parameter

x1 x2

1-11

cl2 k

Value 1.0 2.0 5.0 10.0 10

For the time being, we assume Poisson arrivals for discrete data packets with parameter X2 and for voice data units with parameter X1. The transmission times of data packets are exponentially distributed with mean l/p2 and for voice data with mean l/pi. Buffer capacity is limited to Ic = 10. The resulting CTMC is depicted in Fig. 13.41 and the parameters summarized in Table 13.32. States are represented by pairs (i, j) where i, 0 5 i < k denotes the number of discrete data packet!s in the channel and j, 0 5 j 5 1 indicates the presence of a voice data unit in the channel. Under these assumptions and the model shown in Fig. 13.41, some numerical computations are carried out using the SHARPE software package. First, transient and steady-state results for the rejection probability of data packets

CASE STUDIES OF MARKOV CHAlNS

653

are presented in Fig. 13.42 and for the rejection of voice data in Fig. 13.43. Comparing the results, it can be seen that the rejection probability of voice data very quickly approaches the steady-state value of a little more than 33%. Rejection probability of discrete data is much smaller, on the order of lo-“, and steady-state is also reached in coarser time scale. It was assumed that the system was initially empty.

0

t:::;;~::;;~;;::.:,;;;;I::.:,::,: 0.5 1 Fig. 13.42

1.5

Rejection

3 ! 5 Tirrlc

2.5

probability

for data pack&s.

The channel-idle probability of 66% is approached in a similar time scale as the rejection probability reaches its steady-state for voice data, as can be seen in Fig. 13.44. Often, a GSPN model specification and automated generation of the underlying CTMC is much more convenient and less error prone due to possible state space explosion. The GSPN in Fig. 13.45 is equivalent to the CTMC in Fig. 13.41 [STP96]. The GSPN was also solved using the SHARPE package. The results of Fig. 13.46 illustrate the qualitative and quantitative difference in rejection probabilities of voice vs. discrete data. Note that rejection probabilities increase for voice data as channel capacity is increased. This effect is due to the reduced loss rate of discrete data data packets as a function of increased buffer size. Figure 13.47 shows the impact that the arrivals of discrete data packets have on voice rejection probabilities as compared to rejection of discrete data packets themselves. Both steady-state and transient probabilities are presented. Note that voice data is much more sensitive to an increase in X2. 23.2.4.2 Markov Modulated Poisson Processes Markov processes (MMPP) are introduced to represent correlated

modulated Poisson arrival streams and

654

APPLICATIONS

0.32--

0.3P

0.28-*h I= 3 2 0.26-az

q ____________________

m

a

o Transient

Stationary

0.24--

0.22--

0.2--

: t-

0

I

:

:

:

0.2

Fig. 13.43

;

I

:

:

:

:

0.4

Rejection

I

:

:

:

:

0.6

probability

I

:

:

for voice data units.

0 Transient q _________..______

Fig. 13.44

Channel

idle probability

---.a

Stationary

as a function

:

1 Time

0.8

of time.

CASE STUDIES OF MARKOV CHAINS T Arrival-Voice

655

T Arrival-Data

Data

T Ser-Voice

Fig. 13.45

T Ser-Data

SPN representation of the CTMC in Fig. 13.41.

e Data Packets

Fig. 13.46

Rejection probabilities

as a function of channel capacity k: voice vs. data.

bursty traffic more precisely. An MMPP is a stochastic arrival process with an arrival rate governed by an m-state CTMC [FiMe93]. The arrival rate of an MMPP is assumed to be modulated according to an independent CTMC. The GSPN model of Fig. 13.45 is now modified so as to have two-state MMPP data arrivals as shown in Fig. 13.48 [STP96]. A token moves between the two places MMPPr and MMPP2. When transition Tarrival-datai is enabled and there are fewer than k packets transmitted across the channel, then discrete data is arriving with rate X21. Alternatively, the arrival rate is given by X22. If Xsr differs significantly from X22, bursty arrivals can be modeled this way. Note that the MMPP process is governed by the firing rates of T2r and Tr2, which are given by rates a and b, respectively.

656

APPLICATIONS

+ ____________________ +

e Voice,

*

0

Fig. 13.47

‘-’

2

‘-’

-4

Rejection probabilities

Voice

Stationary

t

= 0.5

O----------------Q

Data Stationary

o--------------------o

Data,

‘-’

t = 0.5

__-..-- ______-da------6 ‘-’ 2

sx3---

__-- *m Q 10

x2

as a function of X2 and time: voice vs. discrete

data.

The results in Fig. 13.49 indicate that for small buffer sizes a Poisson traffic model assumption leads to more pessimistic rejection probability predictions than those based on MMPP assumptions. This effect is quickly reversed for discrete data if channel capacity is increased. For larger capacities there is hardly any difference in rejection probability for voice data, regardless of Poisson- or MMPP-type traffic model. In general, though, correlated or bursty traffic can make a significant difference in performance measures as opposed to merely Poisson-type traffic. But our modeling methods based on GSPN (and SRN), w h’ic h a11ow us to compactly specify and automatically generate

Table 13.33 Parameters to the MMPP-based ISDN model in Fig. 13.48 Parameter

Value

Xl x21 x22

1.0 9.615 0.09615 5.0 10.0 8.0 2.0 10

I*1

P2 a b k

CASE STUDIES OF MARKOV CHAINS

657

T21 n

7’

Arrival-Voice

\

I

I

I

T Ser-Voice

Fig. 13.48

T Ser-Data

A GSPN including an MMPP modulating data packet arrival rates.

G------~--------~...---O

Voice

(M~pp)

t

+ ________----------

----+

Data

(Poisson)

.msm--sT _________--- ---+

Data

(MMPP)

6

Fig. 13.49 MMPP.

Rejection probabilities

+ k

10

as a function of channel capacity k: Poisson vs.

658

APPLICATIONS

and solve large CTMCs and MRMs for steady-state and transient behavior, are capable of handling such non-Poisson traffic scenarios. 13.2.5

ATM

Network

under

Overload

Consider a variant of the ATM network model studied in [WLTV96], consisting of three nodes, N 1, Nz, and N3. Two links, Ni N3 and Nz N3, have been established as shown in Fig. 13.50. Now suppose that the network connection Node 3

Node 1

Node 2

cell arrivals

Fig. 13.50

The [WLTV96] c1ueueing model before rerouting.

between Ni and Ns is down. Before the global routing table is updated, the cells destined to node N3 arriving at node Ni will be rerouted to node Nz (see Fig. 13.51). R ed irected cells from node Nr may be rejected and thus lost if the input queue of node N2 is full. Right after the rerouting starts, the cell loss probability of Ni will overshoot because node N2 is overloaded. In the original paper, the burstiness of the ATM traffic is modeled by a two-state MMPP arrival process. The cell transmission time is modeled by an Erlang-5 distribution. Here we approximate the cell arrival process by a Poisson process and the cell service time by an exponential distribution. We can easily allow MMPP arrivals and Erlangian service time, but the underlying CTMC will become larger. First, we analyze the simple case before the link Node 1

Node 3

X arrivals

I rerouting

Node 2

p-$uM

cell arrivals

Fig. 13.51

I

raster 2

The [WLTV96] queucing model after rerouting.

CASE STUDIES OF MARKOV CHAINS

659

NiN3 breaks down. Here we assume that node Ni and node N2 have the same structure. That is, their buffer sizes and the service rates are the same. Furthermore, we assume that the job arrival rates are the same for the two nodes and that the buffer size at node N3 is large enough to handle all the jobs coming from node Ni and node Nz. Thus node Ni and node Nz can be modeled by a finite-buffer M/M/l/N queue, where N is the buffer size.

Fig. 13.52

The CTMC

for the AI/AI/l/3

queue.

The CTMC and the GSPN models for the node are shown in Fig. 13.52 and Fig. 13.53 respectively. We use SHARPE to find the steady-state probabilities for each state as well as the steady-state cell loss probability before the link NiNa breaks down. Figure 13.54 shows the SHARPE input file of the CTMC q ueue together with their outputs. and the GSPN models of the M/M/1/3 Both models give the same results for the cell loss probability.

AL=-o-i

arr

Fig. 13.53

N

router

The GSPN model for the M/M/l/N

queue.

When the link NiN3 breaks down, the state transition diagram of the subnet can be given as shown in Fig. 13.55. Each state is identified by a pair (i,j) with i,j E (0,. . . ,3}, representing the number of cells in the buffers of node Nr and node Nz. The cell arrival rates are Xi and X2 and the cell transmission rates are ~1 and ~2. States 30, 31, 32, and 33 represent the buffer full states for node Nr, while states 03, 13, 23, and 33 represent the buffer full states for node N2. The loss probability (Ll) of cells destined to node Na going through node Ni is determined by two elements: 1. The probability

that node Ni is full, denoted by pi.

2. The probability that the rerouted cells are dropped because node Nz is full, denoted by d. In short: Lr = pr + (1 - pi)d. The loss probability (Lz) of cells destined to node Na going directly through node N2 is simply the probability that node N2 is full, which is denoted by P2*

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Fig. 13.54 SHARPE input file for the CTMC and GSPN rnodcls of the M/M/ queue together with the results.

In Figs. 13.56, 13.57, and 13.58 the SHARPE input file for the irreducible Markov reward model from Fig. 13.55 is given. From line 1 to line 6, we assign the values to the input parameters, and from lines 10 through 62, the Markov reward model is defined. Lines 10 through 26 describe the CTMC. By using loops in the specification of Markov chain transitions, we can generate the Markov chain automatically. Without the loop functionality, it would be cumbersome to specify the Markov chain manually, especially when the state space gets large. From lines 29 through 45, the reward rates ri,j are assigned to the states (i, j). By default, any state not assigned a reward rate is assumed to have the reward rate 0. In order to analyze the transient behavior of the system, the initial state probabilities are assigned from the values for the steady-state probabilities before the link NrNa breaks down. In other words:

CASE STUDIES OF MARKOV CHAINS

661

fig. 13.55 The CTMC for Fig. 13.52 after rerouting. where ~i,j is the initial state probability for state (i, j) in the transient model. The probability i~~~,iis the steady-state probability of node Ni in state i before the link breaks down. Similarly, the probability 7rs,j is the steady-state probability of node Nz in state j before the link breaks down. The fact we wish to carry out transient analysis of an irreducible CTMC and hence need to assign initial state probabilities is indicated to SHARPE with the keyword readprobs in line 10. The assignment of the initial probabilities is done from line 46 to line 62. Reward rates for computing the average queue length of node Nr are written in a separate file called reward.quellength. In this file, we assign the reward rates to indicate the number of jobs in the first node. The contents of the file are as follows:

In line 65, the keyword include is used to read the file. From lines 67 through 69, we ask for the steady-state expected queue length of Nr. This result is achieved by asking for the expected steady-state reward rate using the keyword exrss. In line 72, we assign the new reward rates so that the steady-state probability that node Ni is full can be calculated. Thereafter we bind the value of pl to p. Similarly, from lines 83 through 121, we assign different reward rates and ask for the steady-state average queue length of

662

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fig. 13.56

fig. 13.57

First

part of input

Second part of input

file for the Markov

file for the Markov

reward

model.

reward model.

CASE STUDIES OF MARKOV CHAINS

f 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 164 156 166 157 168 169 160 161 162 163

663

‘\ * find throughput fox router 2 vax tputx:! exrss (rerouting) * find steady-state drop probability vax d (tPuta2+tputal-tputr2)/tputai

fox rerouted cslJ.s

vax Ll p+(i-PI *d echo steady-stats Closeprobability of cells destined to echo node N3 going through nods Ni expr L1 bind k 3

/

* staxt the transient Loopt ,3, 90 V 3

analysis

bind * pt is transient probability that queue1 is full. sum(i,O,k, tvalus(t;rsToutfng,3,$(i))) pt * p2t is transient probability l;hat qusaue%is full+ sum(i,O,k, tvalue(t;reroutinlT,$(i),3)) P2t * tputa2t is transiant throughput queue2 tputa2t lam2 * ( 1 y sum(i,O,k, tvalua(t;rerouti~,$(i),3))) * tputalt tputait

is transient throughput fox queue1 laml * ( 1 - suan(l,O,k, tvalue(t;seroutiag,3,$Ci)))

* tputr2t tputr2t

is transient throughPut fox router2 mu2 * Cl - sum(i,O,k, tvalue(t;rerouting,$(f),O)))

* dt is transient drop probability for rerouted cells dt CtPutaZt + tputalt - tPutr2tI /tPutalt * Lit is loss probabil.iCy of cells destined to node NS * going through node N1. L&t pt +(S-ptI* dt end end end

fig. 13.58

/ Third

part of input

file for the Markov

reward

model.

node N2, the steady-state probability for node N2 to be full, the throughput of node Nr , node N2, and router 2. Finally we get the cell drop probability for the rerouted cells and the loss probability of cells destined to node Ns going through node Nr. With these steps, we can see the power of the Markov reward model: By just changing the reward rates, we can get different performance measures, such as throughput, average queue length, and rejection probability for the arriving cells. From line 133 to line 163, we perform transient analysis of the model. This result is achieved by using the keyword tvalue. With tvalue, we can get the probability that the system is in state (i, j) at time t. We use the nested sum functions to add up the transient probabilities. From lines 138 through 150, we use the evaluated word notation. An evaluated word is made up of one

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or more occurrences of $(n). Whenever the evaluated word is used, the string $(n) is expanded into an ASCII string whose digits are the number n. For example, in line 138, the state name 3-$(i) includes the states: 3-0, 3-1, 3-2, and 3-3. In Figs. 13.59 and 13.60 the results from the analysis are shown.

Fig. 13.59

Fig. 13.60

Steady state results.

Results of transient analysis.

From the results, we can see that although both router 1 and router 2 have the same cell arrival rate, the steady-state probability for node Nz to be full is more than four times bigger than that of node Ni. This condition occurs because router 2 has two cell sources, one coming from the original cell source, the other from the rerouted cells coming from router 1. However,

665

CASE STUDIES OF HIERARCHICAL MODELS

the loss probability of cells destined to node Na going directly through N2 is smaller than that going through Nr first. From the transient results, we can answer questions such as: l

How long does it take to reach steady state after the line breakdown?

l

What is the loss probability steady state?

at a particular time before the system is in

These are important questions that need to be considered when we make performance and reliability analysis. Figure 13.61 is derived from the transient analysis.

1'0

2'0

Fig. 13.61 Cell loss probability tion 13.2.5.

3'0

4'0

50

60

I 70

t

for the two nodes in the network discussed in Sec-

Problem 13.14 Modify the model of overload described in Section 13.2.5 to allow for two-state MMPP sources and three-stage Erlangian service. You may find it convenient to use GSPN rather than a direct use of CTMC.

13.3

CASE STUDIES

OF HIERARCHICAL

MODELS

We discuss two examples of hierarchical models in this section. Several other hierarchical models are discussed in Chapter 10. For further exposition on hierarchical models, see [MaTr93].

APPLICATIONS

666

Memory1 -ID-(t . . -Ii-o Memory,

Sink Xfj

fig.

13.62

13.3.1

The outer model for a multiprocessor system with cache protocols. A Multiprocessor

with

Different

Cache Strategies

Section 13.1.1 introduces different models for loosely and tightly coupled multiprocessor systems. Now we model multiprocessor systems with different cache coherence protocols. Our treatment is based on [YBL89]. The outer model is a product-form queueing network, some of whose parameters are obtained from several lower level CTMC models. The outer model shown in Fig. 13.62 (where n = 2 CPUs are assumed). The model consists of open and closed job classes where closed job classes capture the flow of normal requests through the system and open job classes model the additional load imposed by the cache. For the queueing network (outer model), the following notation is used: m

Number of memory modules

n

Number of CPUs

PCPU

Service rate at the CPU

pcache Service rate at the cache pbus

Service rate at the bus

Pmem

Service rate at the main memory modules

and for the CTMC cache models (inner models), we use:

CASE STUDIES OF HIERARCHICAL MODELS

667

c Gc c PC

Number of private cache blocks

Cic

Number of instruction

S

Degree of sharing

fuJ

Probability

that data/instructions

are written

1 - fw

Probability

that data/instructions

are read

h

Probability

of a given request to a private cache being a hit

U

Probability

that a previously accessed block has not been modified

d

Probability that a private block, selected for replacement, is modified and needs to be written back

Pit

Probability request

1 - Pit

Probability that a data block is accessed during a memory request; these blocks can be either shared or private

Overall number of blocks in each cache Number of shared cache blocks (blocks shared by all caches)

that

cache blocks

instructions

are accessed during

a memory

The cache behavior is modeled by the jobs of the open job classes. These additional customers induce additional traffic: l

The traffic at the cache due to requests from other caches (loading of missed blocks).

l

The requests from other processors due to state updates.

l

Additional

Fig. 13.63

bus traffic due to invalidation

CTMC for an instruction

signals.

cache block (inner model 1).

If we access blocks in a cache, we have to differentiate between accessing private blocks, shared blocks, and instructions. Private blocks can be owned only by one cache at a time. No other cache is allowed to have a copy of these blocks at the same time. Since it is possible to read and modify private blocks, they have to be written back to the main memory after being modified. In order to keep shared block consistent, a cache coherence protocol needs to be used because shared blocks can be modified by any other processor that also owns this block. Items in the instruction cache can only be read but never be

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modified. Therefore no cache coherence protocol is necessary for instruction blocks. Now we develop and solve three inner models: For an instruction cache block, for a private cache block, and for a shared cache block. A block in the instruction cache can either be not in the cache or valid. If a block is selected for replacement, it is overwritten (no write back is necessary) since these blocks can not be modified. This assumption leads to the CTMC, shown in Fig. 13.63. The transition rates are given by: a=

(1 - fw )Pic Cic

with Ci, = pi,C. obtained:

pcpu,

Steady-state probabilities

b = q~cpu,

of this simple CTMC are easily

(13.11) Since private cache blocks can also be modified, they have to be written back notin

fig. 13.64

CTMC for a private cache block (inner model 2).

after being modified. Modified blocks are called dirty. The corresponding state diagram is shown in Fig. 13.64. The transition rates are given by:

1- fzu b = (l - Pic)(l

- S)~/-+U, PC

l-h c= - 0

CPU,

with CPc = (1 - pic(l - S)C. Steady-state probabilities PC To -

C

a+b+c’

r1

PC -

(a

+ b +a&(b + c) ’

of this CTMC are: PC --

x2

&.

(13.12)

For private and instruction cache blocks no coherence protocol is necessary, while for shared blocks a cache coherence protocol must be used since these blocks can become inconsistent in different caches. One of the first cache

CASE STUDIES OF HIERARCHICAL MODELS

669

coherence protocols found in the literature is Goodman’s write-once scheme [Good83]. When using this protocol, the cache blocks can be in one of five notin

Fig. 13.65 CTMC for shared cache blocks (inner model 3).

1. Notin: the block is not in the local cache. 2. Invalid: the block is in the local cache but not consistent with other caches. 3. Valid: the block is in the local cache and consistent with the main memory. 4. Reserved: the block is in the local cache, consistent with the main memory, and not in any other cache. 5. Dirty: the block is in the local cache, inconsistent with the main memory, and not in any other cache. If a read miss to a shared cache block occurs and a block in another cache exists that is in state dirty, this cache supplies the block as well as writing it back to the main memory. Otherwise (no cache has a dirty copy), each cache with a copy of that block sets the block state to valid. In case of a write miss the block is loaded from memory or, if the block is dirty, the block is loaded from the cache that has a dirty copy (this cache then invalidates its copy). When a write hit to an already dirty or reserved cache block occurs, the write proceeds locally, and if the state is valid, the block is written. In case of a write hit to an invalid cache block, the local state of the cache block is changed. Finally, if a block is loaded into the cache, it can happen that there is not enough space in the cache and a block in the cache is to be selected for replacement. Blocks are replaced in the order invalid, dirty, reserved, and valid. The state diagram of the CTMC for shared cache blocks is shown in Fig. 13.65. Transition rates for this CTMC are:

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670

The processor under consideration generates a read request to a shared block:

l

notin --+ valid :

SC1 - ftu)(l

- 2-k) cLcpu,

Gc

invalid +P valid : If the cache block is in state dirty, reserved, or valid, the read request can proceed without any state change and bus transaction. Another processor generates a read request to a shared block:

l

reserved --+ valid :

Cn -

dirty -+ valid :

Cn -

l)S(l

-

fw)(l

- Pit)

Gc ljs(l

-

fw)(l

Pcpu9

- Pit) Pcpu *

Gc

If the cache block is in the states valid, invalid, or notin, no state change occurs. The processor under consideration generates a write request to a shared block:

l

valid + reserved :

(n -

l)Sfw(l

- Pit)

GC

reserved + dirty : invalid --+ dirty :

Pcpu7

Cn- l)sful(l - pit) Pcpu7 GC Cn -

‘lsfW(l

- Pic)pcpu,

Gc

notin + dirty :

sfW(l

-Pit)

Pcpu*

Gc

If a block is already dirty, the write request proceeds locally without any state change and bus transaction. l

Another processor generates a read request to a shared block: dirty + invalid :

Cn -

l)Sjtu(l

-Pit)

kPU7

Gc

reserved -+ invalid :

(n -

valid + invalid :

(n -

l)Sf~(l

-Pit)

Gc l)Sf~(l

Gc

-Pit)

Pcpu7 PCPU.

671

CASE STUDIES OF HIERARCHICAL MODELS l

A write or read miss in the local cache can result in a page replacement, which leads to the following state transitions: l-h reserved + notin : ccc

cpu >

l-h dirty -+ notin : L__ 0

cpu 7

l-h valid --+ notin : cc1

cpu >

l-h invalid + notin : 0

cpu *

Let: l-h a= -

0

b=

cpu,

s(l

-

fw)(l

- Pit)

I-1 CPU,

C=

s.fw(l

- Pi,)

~ CPU,

Gc

Gc

with C,, = (1 - pic)S . C. Then the steady-state probabilities are given by:

of the CTMC

L

nr =

a+b+c’ a(7rE + 7rT) + (n - l)a(l - 7rF - 7rF) = 7 (n - 1)a + nb + c

$ TT=

(n-l~~~nb+c’

*y = b(” - w - m a+nb+c

’

Now we are ready to parameterize the outer model. To analyze the outer model given in Fig. 13.62, we need the visit ratios for each station and the arrival rates of the open job classes. We see from the following that these parameters are functions of the state probabilities of the three inner models: Visit

Ratio for a Remote Cache i: A cache miss is either supplied by a remote cache that has a dirty copy or by one of the main memory modules if no dirty copy exists. Because the probability of a dirty and valid copy are 1 - (1 - YTT)+~ and 1 - (1 - $)lL-‘, respectively, the visit ratio of the remote cache i is given by: eci =

(1 - J?ic)S(7rr + ry)(l - (1 - 7rF)7L--I) n-l + JlicTi( 1 - (1 - 7rF)n-1) n-l

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APPLICATIONS

Visit

Ratio to Main Memory Module i: The main memory is involved when a cache miss to a private block occurs, a cache miss to a shared block occurs and no other cache has a dirty copy, and a cache miss to instructions occurs and no other cache contains these instructions. Hence the visit ratio is: eMi =

(1 -JIic)(l

- S)(l - h) + (1 -pic)S($ m + pi,7rE(l - ~~)?z-l .

+ -iry)(l - 7rr)n-1

m

Visit

Ratio to the Bus: The bus is used to broadcast invalidations and to transmit missed blocks. Hence the visit ratio is: eB =2(1 - pic)S(7$ + 7$) + 2(1 - psC)(l - S)(l - h) + (l - PSC)Sfw~~ + (1 - psc)(l - S)f,u

+ picrf.

Once we know the visit ratios and the structure of the network, we can compute the routing probabilities, using Eq. (7.5). As a result, we get for the routing probabilities: m

(eC+ Pcachei

eB> +

+proci =

c

j=l

eMj

,

eci

Qi=l,,..,

n,

m eB -

c

eMj

j=l

Pcachei-+bus

=

eci

,

Vi = l,...,

12,

,

Vi = l,...,

n,

m eB -

c

eMj

j=l pbuwproc,

=

eB

= 2

Pb us+memj

Vj = l,...,m.

Next we obtain the arrival rates for each open customer class: Derivation of XC: The possible customers of a local cache are the requests from its owner processor (1 - pit) (1 - S) (1 - h), the loading of missed blocks (1 - pi,-)S(rr + R?) + pier:, the requests from other processors for state updating due to a write request to a shared block (n - 1) (1 pic)Sfzu(~~+~~), and requests from other processors for state updating due to a read request for a reserved block (n - 1) (1 - pit) S( 1 - fW)7rF. Therefore the open customer arrival rate, denoted by XC, is given by: ‘C

=PiPCpUi

((1 -pic)(l

- S)(l

- h) + (1 -P;c)S($

+ PicXE + (n - 1)(1 -

pic)Sf~(K~

+

-

(n

-

l)(l

-

Pic)S(l

f~)$?).

+ ry)

+ 7$)

(13.14)

CASE STUDIES OF HIERARCHICAL MODELS

673

Derivation of Xg: A selected block needs to be written back to main memory only if it is private and modified or shared and dirty. The probability for a shared block to be selected is given by C,, (1 - 7t-r)/C and the probability for a private block to be selected is given by C,,( 1 - $“)/C. The selected block is written back if it is private and modified or shared and dirty. The arrival rate at the bus is, therefore, given by: XB

=piPcpu~n((l

-Pic)(l

-

Cpc(1

-

S)(l 7$“),

-

h)

+

Gc(l

C

-(

+

(1 -

-pic)S7$) qy)

C

7rF .

(13.15)

>

Derivation of XM: There are two possibilities: A shared read results in a miss and one of the remote caches has a dirty copy ( l-pic)S( l- fW) (or+ 7T~)(l-(l-,rr~)n-l), or a write hit to a valid shared block occurs and a write through operation is performed (1 - pic)Sfw + (1 -piJ (1 - S)fWu. The arrival rate at the memory modules is, therefore, given by: AA4 =pi&pui

n((l -pi,)S(l+

(l

-Pic)Sftu

fw)(rE +

(1

-Pic)(l

+ 7ry)(l-

(1 - 7rr)n-‘)

S)fwU).

(13.16) Because the arrival rates XC, Xg, and XM are functions of server utilizations that are computed from the solution of the outer model, the outer model needs to be solved using fixed-point iteration. The solution algorithm is sketched in the following: Solve the inner model for shared/private/instruction cache blocks he steady-state probabilities (see Eqs. (13.11), (13.12), and (13.13)). Determine the arrival rates Xc (Eq. (13.14)), XB (Eq. (13.15)), q. (13.16)) of the open customer classes, assuming the initial value i = 1 and using the steady-state probabilities of the inner model. Solve the outer model using the computed arrival rates to obtain s for the utilizations pZ. The computed utilizations

are used to determine the new arrival

The last two steps are repeated until the results of two successive iteration steps differ less than E. For our example we assume the following parameter values (all time units are msec) : C = 2048 s = 0.1 u = 0.1 d = 0.4

PCPU

/f&ache fibus Pmcm

= 0.25 =

1.0

= 1.0 = 0.25

h = 0.98 fw = 0.3 pi, = 0.1

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674

Table 13.34 Performance measuresfor the cache model with j&us = 1.00 No. of Processors

PCPU

2 3 4 5 6 7 8 9 10 15 20

0.475 0.454 0.438 0.425 0.413 0.402 0.391 0.377 0.363 0.279 0.219

f'bus

0.356 0.426 0.495 0.562 0.619 0.679 0.734 0.785 0.821 0.902 0.902

System Power 0.950 1.278 1.754 2.127 2.478 2.817 3.131 3.395 3.634 4.180 4.374

In Tables 13.34 and 13.35 the performance measures for different mean bus service times are given as a function of the number of processors. Here, the measure system power is the sum of all processor utilizations. In Table 13.34 we see that by increasing the number of processors from 2 to 20, the system power increases by a factor of 4.6. Because the system bottleneck is the bus, it makes more sense to use a faster bus or to use a dual bus instead of increasing the number of processors. In Table 13.35 the results are shown for a faster bus with pbUs = 1.5. As a result, the system power is increased by a factor of 6.5. Table 13.35 Performance measuresfor the cache model with pbUs= 1.50 No. of Processors

PCPU

Pbus

System Power

2 3 4

0.511 0.486 0.470 0.459 0.451 0.445 0.439 0.432 0.427 0.379 0.331

0.255 0.303 0.354 0.404 0.451 0.501 0.549 0.601 0.643 0.817 0.909

1.023 1.458 1.882 2.297 2.707 3.115 3.515 3.892 4.269 5.688 6.614

6 8 9 10 15 20

13.3.2

Performability

of a Multiprocessor

System

We consider a model that was developed by Trivedi, Sathaye, et al.[TSI+96] to help determine the optimal number of processors needed in a multiprocessor system. A Markov reward model is used for this purpose. In this model, it is assumed that all failure events are mutually independent and that a single repair facility is shared by all the processors. Assume that the failure rate

CASE STUDIES OF HIERARCHICAL MODELS

675

of each processor is o. A processor fault is covered with probability c and is not covered with probability 1 - c. Subsequent to a covered fault, the system comes up in a degraded mode after a brief reconfiguration delay, while after an uncovered fault, a longer reboot action is required. The reconfiguration times are assumed to be exponentially distributed with mean l/y, the reboot times are exponentially distributed with mean l/S, and the repair times are exponentially distributed with mean l/,8. It is also assumed that no other event can take place during a reconfiguration or a reboot phase. The CTMC modeling the failure/repair behavior of this system is shown in Fig. 13.66. In state i, 1 < i 5 n, the system is up with i processors functioning, and n - i processors are waiting for repair. In states (c,-i), i = 0, e. . , n - 2, the system is undergoing a reconfiguration. In states (b+i), i = 0,. . . , n - 2, the system is being rebooted. In state 0, the system is down waiting for all processors to be repaired.

Fig. 13 .66

CTMC for computing the performability

of a multiprocessor

system.

The reward rate in a state with i processors functioning properly correspond to some measure of performance in that configuration. Here the loss probability is used as a performance measure. An M/M/i/b queueing model is used to describe the performance of the multiprocessor system, which is represented by the PFQN shown in Fig. 13.67. This model contains two stations. station mp is the processor station with i processors; it has type ms for multiple servers, each server having service rate ,L The other station is source, which represents the job source with rate X. Because there is a limited number b of buffers available for queueing the jobs, the closed product-form network with a fixed number b of jobs is chosen. The loss probability can be obtained via the throughput tput of station mp. Note that we have used a two-level hierarchical model. The lower level (inner model) is the PFQN (Fig. 13.67), while the outer model is the CTMC (Fig. 13.66). For state i of the CTMC, the PFQN with i processors is evaluated, and the resulting loss probability is used as a reward rate in state i. Then the expected steady-state reward rate is the overall performability measure of

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APPLICATIONS

Fig. 13.67 PFQN model of the M/M/i/b

queue.

the system, given by EXRSS = 2

rj7ri,

i=l

where 7ri is the steady-state probability of state i, and ri is the loss probability of station mp. We consider just the operational state i because in the down states ci, bi and 0, the loss probability (reward rate) is 1. arr

buffer

Fig. 13.68

request

serving

service

GSPN model of the M/M/i/b

proc

queue.

It is also possible to model the lower level model using the GSPN shown in Fig. 13.68, The initial number nproc of tokens in place proc means that there are nproc processors available. When a new job arrives in place bujFer, a token is taken from place proc. Jobs arrive at the system when transition arr fires. There is a limitation for new jobs entering the system caused by the inhibitor arc from place bufler to transition arr. Thus arr can only fire when the system is not already full. There can be only b jobs in the system altogether, nproc being served (in place serving) and b- nproc in place buffer. The firing rates are X for transition arr and k,x for transition service. Here k is the number of tokens in place serving; the notation for this markingdependent firing rate and in Fig.13.68 is p#. The expected reward rate in the steady state is called the totd loss probability (TLP). It is defined as the fraction of jobs rejected either because the buffer is full or the system is down. A possible SHARPE input file for this hierarchical performability model is shown in Fig. 13.69. Descriptions of the lines of the SHARPE input file are as follows: Line 3-9: Assign the values to the input parameters.

CASE STUDIES OF HIERARCHICAL MODELS

Fig. 13.69 Input file for multiprocessor performability

677

model.

Line 12-37: Define the GSPN model; the model has two parameters, the number of processors nproc and the job arrival rate lambda. Line 39-40: Define a function to use for reward rates. Line 42-48: Specification of a Markov model for nproc = 1 processor. Line 46-47: Assign reward rates to all states. Line 50-63: Specification of a Markov model for nproc = 2 processors. Line 59-62: Assign reward rates to all states. Line 65: Input file with specification of a Markov model for 3 processors. Line 66: Input file with specification of a Markov model for 4 processors. Line 67: Define models for more processors.

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APPLICATIONS

Line 69: Let the arrival rate L vary from 50 to 200 by increments of 50. Line 70-75: Get the TLP for each number of processors. In Fig. 13.70 the total loss probability and system unavailability (TLP with X = 0) are plotted as functions of the number of processors. The difference between the unavailability and the total loss probability is caused only by limited buffer size b.

0.1

0.01

0.001

0.0001

0.00001

5

4

Number

Fig. 13.70

6

of processors

TLP and unavailability.

B

1'0

Glossary

place Pi in marking

#Pi, 4 VPZ 44

factor for computing

the shadow

argmin

function

the index of the minimum

ABA

asymptotic

bounds

AMVA

asymmetric

MVA

APU

associate processing

ASCAT

asymmetric

ASPA

average sub-system

ASUM

asymmetric

ATM

asynchronous

bi,3

probability that a job at node i leaves the node after the jth phase

B

normalized

number

m

mean service time of the jobs at the ith node

blocking

BN

BCMP

of tokens at a particular

network

that returns

service time value

analysis unit

SCAT population

algorithm

SUM transfer

blocking

mode

factor

factor

queueing network with several job classes, different queueing strategies, and generally distributed service times, named after Baskett, Chandy, Muntz, and Palacios

BFS

Belch, Fleischmann

BJB

balanced

job bounds

and Schreppel analysis

method

680

GLOSSARY

BOTT

bottapprox

method

CAz

coefficient

of variation

EA

Coefficient

of variation

CBz

coefficient

of variation

CBCO

coefficient the closing

of variation method

4.9

capacity function pleted by station normalized function

that

Cl

index

corr

correction

ck

Cox

distribution

m-4

linear

cost

for

of the

batch

of the

service

closed

deviation

time

time time

when

or coefficient

up to the

job

interarrival

applying

i; the number of jobs comare j jobs in the queue

of node i if there

rounds

time

interarrival

of the service

standard

ceiling

next

of variation

integer

classes

factor with

Ic phases

function

CCNC

coalesce

CDF

cumulative

CPU

central

processing

CSMA/CD

carrier

sense

CTMC

continuous

49 driv

function

context

of the

computation

of normalizing

distribution

constants

function unit

multiple time

access

Markov

of the jobs

a job executes operation deterministic

k-s(K)

change of contribution s job is removed from

D+

output

D-

input

DA

diffusion

DAC

distribution

DES

discrete-event

DTMC

discrete

detection

network to set up or perform

code

D

collision

chain

in the

driver

with

an I/O

distribution in the SCAT the network

algorithm

if a class-

arcs arcs approximation analysis simulation

time

stopping

by chain

Markov

chain

condition

e

Vector

e,

visit ratio; node, also

the mean number of visits of a job known as the relative arrival rate

visit ratio method

of the

of the

Visit

ratios

closing

e = (ea, er, . . . , ejV> to the

applying the

ith

681

GLOSSARY

number ratio

of visits

at node

ecz(k) ERG

effective

capacity

function

El,

Erlang

ET.

set of pairs (j, s) that can be reached a finite number of steps

f(CA f

extended

7 CB,

t> dFx

f&d

=

(x) dr

that

density

function

probability

density

function

needed

function

F, F,(k)

functions corresponding of the ith node

Fw (K)

contribution population

5 z)

that

for

miss

= P(X

overlap

probability

floor

Fx(x)

k phases

with

of all I/OS

functions

fk?-

ratio

the

rounds

from

with

a primary

fixed-point

to the

task

iteration

next

lower

to the state

of class-r

cumulative

distribution

FCFS

first-come-first-served

FDDI

fiber

FES

flow

FTCS

symposium

g@>

core

rk4 G

gamma

distribution

general constant

distribution,

batch

integer

jobs

at the

7rTT2 (k,)

probabilities ith

node

for

a given

function

policy

distributed

data

equivalent

server on fault

function

interface method tolerant

of the

computing

summation

also

method

used

for

the

G(K)

normalization work

constant

of a single

G(K)

normalization work

constant

of a multiclass

GKLB

correction

Gn(k)

auxilary auxiliary

(i, r) in

K

full

(k)

pair

F, = 1 - H,

function

FB

Gn

called

approximation

fraction (s;

Q, also

graph

distribution

Kulbatzki

d

01

fx

reachability

i in chain

factor function function

class

normalization

queueing

net-

queueing

net-

Kramer/Langenbach-Belz for for

G;‘(k)

normalization constant from the network

Gkj (k)

normalization node i removed

computing computing

G(K) G(K)

of the network

constant of the multiclass from the network

with

node

removed

network

with

GLOSSARY

GE

Gaussian

elimination

GI

general

distribution

Great SPN

analysis

package

GREEDY

special

case

algorithm with for

of the

minimum

Taksar,

and

GTH

Grassmann,

GUI

graphical

h

function

used

hz

hit

of the

h(P7 4

factor for computing metatos approximation

H/c HW

hyperexponential

user

ratio

entropy

non-preemptive

I/O

input/output

IMMD

IV

i+]

=[p;,

pL;t] interval1

of the

integrated

services

iSPN

integrated Petri nets

environment

k,

number

of the

executes

K-

number

of jobs

K*

overall

number

number population

strategy

rates

machines

and

data

modeling

ith

node

rth

class

using

stochastic

at the

ith

node

code of jobs in a closed of jobs in the system of the of jobs

of jobs in vector

the

rth

class

in the

network,

in the fictious

various

classes,

known

mean

number

in the

as the

an arriving

k

system

mean number synchronization

of parallel-sequential

mean

of jobs

at node

used

network

number of class-s jobs at node i that job sees given the network population of jobs

also

network

mean class-r

number

process-

network for

in the

kern

number number

KS

Cos-

network

constant for the

(K1,...,Kn)

service

digital

at the

of jobs

K

K=

of the

k phases

queueing

mathematical

of jobs

number job

case

interval1

ISDN

context

with

head-of-line

server

kern

in the

distribution

= [i-,

state

analysis

WGI,~,~

infinite

. . .,kN)

algorithm

cache

IS

k2,

nets

policy

Heyman

in bottleneck

for

(kl,

batch

times

Petri

interface

institute ing IV

k 27‘

interarrival

stochastic

function

HOL

I Pn

independent

analyzing

jobs

i

with

fork-join

683

GLOSSARY

zr -* K~R*

mean

number

overall network

-* K

of class-r

number

of class-R*

mean number network

x

throughput

$0)

initial ysis)

value

arrival

rate

of jobs

of jobs

from

of the

primary

overall

arrival rate

of batches

rate

arrival

rate

of class-r

open

(used

in bottleneck

anal-

to the

ith

at node

at the

ith

from

in case

node

i to node

throughput

bound,

overall

bound,

of par-

node

of the

pessimistic

node

0

class tasks

throughput, the rate at which jobs serviced and leave the ith node of the

fictious

in the pseudo

task

jobs

i in the

node

outside

of jobs

i

at node

of the open asynchronous

arrival

optimistic

fictious

network

at node

largest

possible

throughput

lowest

possible

throughput

rth

j class

estimation

Lagrange Lagrange

function multipliers

load

Lx

Laplace transform form of the CDF

factor

with objective yl and y2

of open

classes

LAN

local

area

local

balance

property

LBANC

local

balance

algorithm

LBPS

last

LCFS

last-come-last-served

i, class

-R*

X and

two

pdf:

to station

i

Laplace-Stieltjes

trans-

network

batch

Little’s

function

regarding

of the

LB

processor

theorem,

for

normalizing

constants

sharing

also

known

LST

Laplace-Stieltjes

LT

Laplace

Pa

service

rate

of the

PLO0

service

rate

of the closing

as Little’s

law

transform

transform jobs

at the node

ith

node

when

applying

the

ing method Par

are

throughput

likelihood

L

theorem

jobs

for the throughput

throughput

Little’s

at node

at the ith

maximum throughput allel processing with

(4

jobs

service

rate

of the

ith

node

for jobs

of the

rth

class

clos-

684

GLOSSARY

vector of service rates rn() E M denotes the initial number

of parallel

function numbers

marking

of a PN

servers at the ith node

that delivers

the maximum

value of a set of

M

exponential

distribution

Kr

mean number of customers of class-i who arrive during the waiting time of the tagged customer and receive service before him

M+M

Markov

M(t)

time-averaged

MB

minimum

batch policy

MEM

maximum

entropy

MMPP

Markov

MOSEL

modeling,

specification

and evaluation

language

MOSES

modeling,

specification

and evaluation

system

MRM

Markov

MTTA

mean time to absorption

MTTF

mean time to failure

MTTR

mean time to repair

MVA

mean value analysis

implies

Markov behavior

of the CTMC

method

modulated

reward

property

Poisson process

model

MVALDMX

mean value analysis load dependent

44

vector of the state probabilities

44

conditional

Y

vector of the state probabilities chain (DTMC) limiting

throughput

of node i with k jobs of a discrete time Markov

state probabilities

nCDm

factor of the Cosmetatos

N

product-form number

mixed

at time n

approximation

network

of class-r jobs at node i in the shadow node

mean number of customers of class-i found in the queue by the tagged (priority T) customer who receive service before him

NPFQN

non-product-form

OP

index for open job classes

8

estimate

of 8

ODE

ordinary

differential

queueing

equation

network

685

GLOSSARY

One-limited

service

station arrives

transmits

probability

#)

7rp

probability

r!J

individual

n;(k)

marginal ki = k

a r’j

that all servers are occupied state for all g C G

probability jobs

unconditional

(4

sN>

@l,

that the ith node contains exactly

of node i of a non-lossy

system

state probabilities

probability

rk

the token

of loss

state probability

,W)

at most one frame when

of the number

state probability

of jobs in the system

of a network

with multiple

w

state probabilitiy

7i

derivative

Po,js

probability in an open network that a job from outside enters the jth node as a job of the sth class

PoLi

probability that a job entering first enters the jth node

Pzo

probability that a job leaves the network pleting service at node i

P,,

probability that a job is transferred to the jth node after service completion at the ith node (routing probability)

Pij (t)

simplified CTMCs

Pv (% 4

transition probability of the CTMC i to state j during [u, v)

p’“‘(k a3

probability that the Markov chain transits from state i at time k to state j at time 1 in exactly n = 1 - k steps

7 Z)

vector at any instant

job classes

of the transition

transition

of time u

probabilities

the network

probabilities

from outside just after com-

for time-homogeneous to travel from state

probability in an open network that a job of the rth class leaves the network after having been serviced at the ith node P arkIs

probability that a job of the rth class at node i is transferred to the sth class at node j (routing probability) time-average

accumulated

reward

distribution of the accumulated reward time [0, t), also called performability

over a finite

correction factor for computing the shadow service time in the case of a network with class switching and mixed priorities

Pdf

probability

density function

GLOSSARY

pmf p-i?@(x)

probability

P

places of a Petri net

p7Tl

waiting

I?

derivative

of the matrix

P

stochastic

transition

P(u, 4

matrix

of the transition

p(n)

matrix

of the n-step transition

PANDA

Petri net analysis and design assistant

PASTA

Poisson arrival see time averages theorem

PEPSY

performance

evaluation

PFQN

product-form

queueing

PN

Petri net

w4 PRIOMVA

priority

mass function of a class-r job with respect to a quota node

CDF of the standard

normal

probability

scaling function extended works

of the transition probabilities probabilities

and prediction

system

network

of the extended

SCAT

MVA for the analysis of priority

extension works

PRS

preemptive

PS

processor

qr (t>

priority

Q Q

queue length

6

mean queue length

&

mean queue length

a

mean number of tasks waiting tion in queue Q2.

QNAP

probabilities

matrix

PRIOSUM

aQbatch

distribution

of the summation

method

resume queueing

strategy

algorithm queueing to priority

net-

sharing of class-r at time t

infinitesimal

generator

matrix

for batches for the join synchroniza-

mean queue length

of class-r jobs at the ith node

mean queue length system

for individual

queueing

net-

network

customers

in a batch

analysis package

utilization pbott

utilization

of the bottleneck

Pi

utilization

of node i

P,r (WI

utilization

of node i by class-r jobs

pk

specific utilization of an asymmetric

node

needed for calculating non-lossy system

the utilization

687

GLOSSARY

Pit

utilization

ra

reward

of the kth server of a lossy system rate assigned to state i

starting

priority

reduced

reachability

reachability number

of class-r graph

set

of priority

classes

mean remaining service time of a busy server, also called mean residual life RECAL

recursion

RESQ

research queueing

RPS

rotational

RR

round robin

a$ = var(X)

variance of X or second central virtual

by chain algorithm network

position

package

sensing system moment

service time of class-r jobs at quota node i

mean contention

time

Sa,r

service time of class-r jobs at node i

ga,r

service time of class-r jobs at shadow node i

*

service rate in chain Q at node i

% sa4.d

load-dependent time

S12

mean latency time

SSZ

mean seek time

Sta

mean transfer

Sa s

= =

(Ical, (Sl,...,SN)

* * * 7 ICaR)

mean service time or normal

time

state of the ith node overall state of the network synchronization

SB

service

station

with multiple

classes

overhead

balance property

SCAT

self correcting

SHARPE

symbolic hierarchical evaluator

SIR0

service in random

SMP

semi-Markov

SOR

successive over relaxation

SPN

stochastic

Petri net

SPNP

stochastic

Petri net package

SQL

structured

query language

SRN

stochastic

reward

approximation automated order

process

net

technique reliability

performance

688

GLOSSARY

SUM

summation

7

tangible

markings

T

response

time

T

mean

%

mean response time join synchronization

method

response

time of parallel-sequential

mean

response

time

at node

mean

response

time

of class-r

mean

join

optimistic

bound,

lowest

possible

mean

bound,

greatest

mean

response

rotation

tool

TLP

total

TLP(t)

transient

TPM

transition

time

possible

the join

mean

of a purely

time

of

task

belonging

to

time command

language/tool

kit

loss probability expected

total

probability

loss probability matrix

2 blocking

blocking before service, service blocking, tion blocking, or immediate blocking

3 blocking

repetitive job

Upper

service,

executes

number

time

limit

a purely

of a frame

Type

u

time

of a frame

blocking after service, ing, or non-immediate

context

time

response

job

1 blocking

Type

synchro-

response

sequential

Type

user

i

time

time

mean response sequential job

Tcl/Tk

at node

for

target

transfer

jobs

i waiting

mean

Tt

fork-

i

by task QZ

pessimistic

delay

with

time

mean time spent nization in queue

Td

jobs

tirne time

transfer blocking

blocking,

or rejection

user

production

block-

communica-

blocking

code

of chains limit within system

V

vanishing

w

relaxation

WO

optimal

Wo”

approximation

WT,k

factor

W

waiting

jobs

need

to be served

in a real

markings parameter relaxation

for

parameter of the optimal

computing

time

the

shadow

relaxation service

parameter time

GLOSSARY

W(i, d

weighting

W

mean waiting

ml

mean remaining

function

of the extended

689

SCAT algorithm

time service time of a job in service

m

mean waiting

time at node i

war

mean waiting

time of class-r jobs at node i

WV

mean waiting class-r

time of an arriving

WAN

wide area network

WEIRDP

queueing strategy where the first job in the queue is assigned to the pth part of the processor and the rest of the jobs are assigned to the 1 -pth part of the processor

WinPEPSY

Windows SY

WIP

mean number progress)

ET

set of jobs that cannot be preempted

X ave

average relative the network

2%

relative

Xmax

maximum

Xsum

sum of the relative

Xt

: t E T

version of the queueing of workpieces

utilization

customer

network

package PEP-

in the system

(work

in

by class-r jobs

of the individual

nodes in

utilization relative

family of random

utilization utilizations variables

I?

=E[X],

X”

= E[X”],

nth moment

XPEPSY

X-version

of the queueing

w>

accumulated

& (4

number of possibilities to distribute nodes (including the join queue QZ)

mean value or expected

instantaneous number

of priority

reward

reward

network

value

package PEPSY

in the finite time horizon

tasks to n, + 1

rate at time t

of states in the fork-join

[0, t)

sub-network

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Index

A ABA, 411 Absorbing state, 46 Absorbing marking, 84 Accumulated reward, 66 Aggregation step, 168 Algorithms ERG, 92 AMVA, 535 ASPA, 500 asymmetric SCAT, 536 BFS, 559 BOTT, 405 CCNC, 312 convolution, 313 Courtois’s approximate method, DAC, 312 FES, 368 Gaussian elimination, 144 Grassmann, 119 LBANC, 311 MVALDMX, 352 RECAL, 311 RECAL method, 360 SCAT, 385 core algorithm, 386 extended, 393 multiple server, 389 single server, 385

153

SUM, 397 multiclass network, 403 single class networks, 400 Takahashi, 170 Allen-Cunneen, 443 Allen-Cunneen approximation, 229-230, 234 AMVA, 535 Aperiodic, 47 Approximate lumpability, 169 Approximation algorithms, 573 Approximation networks, 379 APU, 492, 623 Arc multiplicity, 87 marking-dependent, 90 Arcs, 84 Argmin, 405 Arrival overall rate, 266 process, 214 rate, 266 theorem, 326, 343, 473 ASCAT, 536 Associate processing unit, 492 Associated processor model, 624 ASUM, 534 Asymmetric MVA, 535 nodes, 533 queueing systems, 250 SCAT, 536 719

720

INDEX

SUM, 534 system, 248, 250 GI/G/m, 249 M/M/m, 249 ATM networks, 658 ATM, 33

B Bard-Schweitzer approximation, 380 Batch service, 258 Batch size, 258 BCMP theorem, 313 BCMP mixed networks, 352 theorem, 300-301 Version 1, 303 Version 2, 303 BFS method, 559 Binomial coefficient, 297 Birth rate, 105 Birth-death process, 105, 218, 221 BJB, 414 Blocking after service, 548 before service, 548 factor, 521 network, 547, 549 probabilities, 553 rejection, 549 repetitive, 548 types, 548 BOTT, 405, 534 Bottapprox method, 405, 463 multiclass, 408 Bottleneck analysis, 406, 427-428 Bounds analysis, 410 analysis asymptotic, 410-411 balanced job, 410, 414 lower bound, 233, 379 optimistic, 410 pessimistic, 410 upper Boxma Cohen and Brumelle,

bound,

229,

Huffels 233

formula,

233,

379

235

C Cache coherence coherence write-once, miss, 603 strategies,

protocol, protocol 669 666

667-668

Caches, 603 CCNC algorithm, 312 CDF, 7, 9, 14, 26, 36 conditional, 36 Cell loss probability, 658 Central limit theorem, 24 Central-server, 263 model, 263, 331, 511 Chain, 36 concept, 623 Chapman-Kolmogorov equation, 39, 50 Cheapernet, 610 Class switching, 295, 326, 337, 482, 621, 623 mixed priorities, 483 Client server system, 646 Client server systems, 607 Closed network single class, 313 Closed queueing network, 274 Closed tandem network cyclic queue, 222 Closing method, 464 Coefficient of variation, 9, 11, 13-14, 16, 20, 259 Communication systems, 608 Composite node, 452 Concept of chains, 295 Conditional probability, 22 Confidence interval, 29 Conservation law, 243 Contention time, 502 Continuous parameter process, 36 Continuous random variable, 7, 30-31 Convolution, 33 algorithm, 311, 313, 564, 573 method, 317 operator, 313 Correction factor, 229 Correlation coefficient, 24 Cosmetatos approximation, 231, 235, 241 Cost function, 558 linear, 558 non-linear, 559 Courtois’s approximate method, 153 Covariance, 23 Coverage factor, 61 Cox-m, 454 cox-2, 453 Crommelin approximation, 232 CSMA/CD network, 608 CSPL, 579 CTMC, 49, 253, 274, 276, 279, 311, 571 ergodic, 53 state transition diagram, 250 time-homogeneous, Cumulative distribution

49 function,

7

INDEX Cumulative

measure,

178

D DA, 423 DAC algorithm, 312 Death rate, 105 Decision variables, 558 Decomposition method, 439 of Chylla, 442 of Gelenbe, 441 of Kuhn, 442 of Pujolle, 441 of Whitt, 441 Degree of multiprogramming, 263 DES, 1, 59, 573 Diffusion approximation, 423 Disaggregation step, 168 Discipline dynamic priorities, 211 FCFS, 211 IS, 211 LBPS, 307 LCFS, 211 preemption, 211 PS, 211 queueing discipline, 242 RR, 211 SIRO, 211 static priorities, 211 WEIRDP, 307 Discrete random variable, 5, 20, 25 Discrete-event simulation, 571, 573 Discrete-parameter process, 36 Discrete-time Markov chain, 38 Distribution arbitrary normal, 10 branching Erlang, 17 cox, 17, 210 deterministic, 210 Erlang, 210 Erlang-lc, 13 Erlang-r, 33 Erlang-2, 276 exponential, 10, 210, 216, 224 function, 7 gamma, 15 general independent, 2 10 general, 2 10 generalized Erlang, 16 hyperexponential, 12, 210 hypoexponential, 14 k phases, 15 r phases, 34 non-exponential, 423 normal, 9 of sums, 31 waiting time, 217, 227

Weibull, DTMC,

38,

19 296,

571

Embedding techniques, Entropy, 43 1 function, 43 1 Equilibrium, 274 probability vector, Equivalent network, ERG algorithm, 92

ERG,

721

106

53 551

98

Ergodic chains, 326 Ergodic CTMC, 53 Ergodic Markov chain, 47 Ergodic, 47, 54, 274, 285 Erlang-m, 454 Estimate, 28 Estimator, 28 Ethernet, 607 Expectation, 20 Expected instantaneous reward rate, Expected value, 6, 8 Extended BOTT, 463 Extended MVA, 470 Extended reachability graph, 86 Extended SUM, 463

68

F Fast state, 191 FB, 258 FDDI, 610 FES method, 368 multiple nodes, 373 FES node, 368 Fire, 84 Firing rate marking-dependent, 90 Firing time, 86 Fission-fusion, 518 Fixed-point iteration, 403 Flexible production systems, 630 Flow, 441 Flow-equivalent server method, 311, 499 Fokker-Planck equation, 423 Fork-join systems, 517 Full batch policy, 258

G Gauss Seidel, 594 Gauss-Seidel iteration, 140 Gaussian elimination, 144 Generator matrix, 277 GI/G/l system, 229

368,

722

INDEX Kulbatzki, 234 approximation, formula, 234

GI/G/m system, 233 GI/M/l system, 225 GI/M/m system, 228 Global balance, 274 equations, 54, 274, 277, 279, 290 Gordon/Newell theorem, 288, 292 Grassmann algorithm, 119 GREEDY policy, 258 GSPN, 86 Guards, 89

L Lagrange function, 565 multipliers, 560 LAN, 607-608 Laplace transform, 26-27, 301 Laplace-Stieltjes transform, 26, 216, 224-225 Latency time, 502 LBANC, 311 LBPS, 307 Likelihood function, 29 Linear cost constraint, 560 Little’s theorem, 213, 215, 218, 221, 239, 252, 254, 271, 326 Load-dependent, 301 Local balance, 277-278 conditions, 301 equations, 278-279, 283 property, 281, 283 Loss system, 250, 255 LST, 216 Lumpable, 160 approximate, 169 pairwise, 169

H Heavy traffic approximation, 245 Hessenberg matrix, 108 Heterogeneous multiple servers, 248 Hierarchical models, 571, 665

I I/O subsystems, 501 Immediate transition, 86 Impulse reward, 66 Infinitesimal generator matrix, Inhibitor arcs, 87 Initial marking, 84 Initial probability vector, 41 Input arc, 84 Input place, 84 Instantaneous measure, 178 Instantaneous reward rate, 66 Internal concurrency, 506 Irreducible, 46, 54 ISDN channel, 650 ISPN, 585

52

M

J Jackson’s method, 467, 537 Jackson’s theorem, 284 Jacobi method, 137 Jensen’s method, 184 Joint cumulative distribution function, Joint distribution function, 20 Joint probability density function, 36 Joint probability mass function, 20

230

36

K Kendall’s notation, 210-211 Kimura, 443 approximation, 230, 235 Kingman, 233 Kingman-Kijllerstriim approximation, 234 Kolmogorov’s backward equation, 51 Kolmogorov’s forward equation, 51 Kramer/Langenbach-Belz, 234, 443 Kramer/Langenbach-Belz formula, 229 &h-order statistic, 30

M/G/l system, 223 M/G/m system, 230 M/M/o0 system, 217 M/M/l system, 214 M/M/l/K system, 219 M/M/m system, 218 Machine repairman model, 221, 263 Macro state, 167 Marchal, 233, 444 Marginal probabilities, 272, 389 Marie’s method, 452 Marking, 84 dependent arc multiplicity, 90 dependent firing rate, 90 tangible, 86 vanishing, 86 Markov chain, 37 chain ergodic, 47 process, 36 property, 36, 49 reward model, 56, 64 Martin’s approximation, 230 Master slave model, 623

INDEX Matrix equation, 274 Maximization, 558 Maximum entropy method, 430 closed networks, 433 open networks, 431 Maximum-likelihood estimation, 29 MB, 258 Mean number of jobs, 219, 271, 273 of visits, 266, 268 Mean queue length, 219, 225, 271 Mean recurrence time, 47, 54 Mean remaining service time, 239, 243 Mean residual life, 223 Mean response time, 219, 225, 271, 317 Mean time to absorption, 66 to failure, 61 to repair, 62 Mean value analysis, 311, 326, 573 Mean value, 6, 8 Mean waiting time, 219, 223, 271 MEM, 430 Memory constraints, 498, 504 requirements, 382 Memoryless property, 37, 224 Merging, 440 Met hod of moments, 28 of shadow server, 478 of simultaneous displacement, 137 Minimization, 558 Minimum batch policy, 258 MMPP, 653, 658 Moment, 26, 227 nth, 6, 9 central, 7, 9 second central, 7 Monoprocessor model, 621 Monotonicity, 398 MOSEL, 588 MOSES, 489, 588 MRM, 56, 64 instantaneous reward rate, 66 MTTA, 66 MTTF, 61 MTTR, 62 Multiclass closed networks, 320 Multilevel models, 571 Multiple job classes, 267, 271 Multiple servers, 270 Multiprocessor system, 603 loosely coupled, 606 tightly coupled, 603, 666 Multiprogramming system, 263 MVA, 326, 379, 405, 535

MVALDMX

algorithm,

723

352

N Network BCMP, 295 closed, 263, 266, 288, 297, 348 load-dependent service, 347 mixed, 267, 343 BCMP, 352 multiclass, 267, 271 closed, 320 open, 266, 297 product-form, 281, 306 separable, 281 single class, 265, 268 with blocking, 547 Networks approximation, 379 product-form, 379 Newton-Raphson method, 404 Node types -/G/l-FCFS HOL, 495 -/G/l-FCFS PRS, 497 -/G/m-FCFS HOL, 495 -/G/m-FCFS PRS, 497 -/M/l-FCFS HOL, 471, 494 -/M/l-FCFS PRS, 471, 496 -/M/l-LBPS, 307 -/M/l-SIRO, 306 -/M/l-WEIRDP, 307 -/M/m-FCFS HOL, 495 -/M/m-FCFS PRS, 496 product-form, 282 Node, 263 Nominal service time, 352 Non-exponential distribution, 423 Non-lossy system, 251 asymmetric, 252 Non-product-form networks, 421 Normalization condition, 269, 271, 278, 280 constant, 281, 289, 301, 311, 313-314, 320, 360 Normalized blocking factor, 521 Normalized speedup, 521 Normalized standard deviation, 7 Normalized synchronization overhead, 521 Normalizing condition, 255 Norton’s theorem, 368 NPFQN, 3 n-step recurrence probability, 47 transition probabilities, 39

724

INDEX

Number

of jobs,

212-213,

30J

0 ODE, 189 One-limited service, One-step transition Optimization, 557 dynamic, 557 static, 557 Ordinary differential Output arc, 84 Overall throughput,

614 probabilities,

equation,

38

189

270, 273

P Pairwise lumpable, 169 Parallel processing, 507 with asynchronous tasks, 507 PASTA theorem, 343 Pdf, 8-9, 11, 13-14, 16, 19, 33, 36 PEPSY, 572 Percentiles, 73 Performability, 66, 68, 674 Performance evaluation, 620 measures, 212, 268, 277 Periodic, 47 Peripheral devices, 263 Petri net, 84 PFQN, 3, 273 Place, 84 Pmf, 5 PN, 84-85 states, 84 Poisson arrival streams, 301 process, 11, 214 Pollaczek-Khintchine formula, 224, 229 transform equation, 224 Polling, 113 asymmetric limited service, 114 exhaustive-service systems, 114 gated-service systems, 114 single-service systems, 114 symmetric limited service, 114 Population vector, 268, 300 Positive recurrent, 47 Power method, 133 Preemption, 242 Priority, 88, 621 class, 237 fixed, 244 function, 244, 246 networks, 470 queueing, 237 system, 237

static, 248 strategy mixed, 623 system, 244, 248 time dependent, 243 Private cache blocks, 668 Probability n-step recurrence, 47 density function, 8 generating function, 25 marginal, 269, 272, 315 mass function, 5 geometric, 214 of loss, 251, 256 one-step transition, 38 routing, 265-268, 279 steady-state, 214, 218, 221, 227, 268 transition, 49 unconditional state, 52 vector, 212 equilibrium, 53 steady-state, 53 waiting, 219, 231 Product-form, 283 approximation, 429 expression, 289 network, 278, 311, 379 node types, 282 queueing network, 398 multiclass, 399 single, 398 solution, 283, 311 Property local balance, 283 M =+ M, 283 Markov, 36, 49 memoryless, 37, 216 station-balance, 283 Pujolle method, 467

Q Queue length, 213 Queueing network product-form, 571 Queueing discipline, 210-211, 301 network, 263 closed, 279 closed tandem, 263 non-product-form, 571 system asymmetric, 250 symmetric, 250 systems elementary, 210 Queuing network

INDEX

Quota

central-server, nodes,

594 490

Race model, 86 Random selection, 250 Random variable Bernoulli, 5 binomial, 6, 26 Erlang, 20, 27 exponential, 20, 27 gamma, 20, 27 geometric, 6, 26 hyperexponential, 20, 27 hypoexponential, 20, 27 Poisson, 6, 26 statistically independent, 20 sum, 23 Random variables continuous, 22 Randomization, 104 Reachability graph, 98 Reachability set, 84 Reachable, 46, 54 RECAL algorithm, 311 method, 360 Recurrent non-null, 47 Recurrent null, 47 Recurrent state, 47 Recurrent positive, 47 Relative arrival rate, 266 Relative utilization, 266, 411 Relaxation parameter, 140 Remaining service time, 230 Repetitive blocking, 548 Residual state holding time, 55 Response time, 213, 217, 219 RESQ, 517 Reward rate, 66 expected instantaneous, 68 specification, 90 RG, 91 Robustness, 448 Routing matrix, 296, 298 probability, 265 load-dependent, 307, 326

RS, 84 S SCAT algorithm, 536 SCAT, 385 core algorithm, 386 extended algorithm,

393

multiple server node algorithm, 389 single server node algorithm, 385 Seek time, 502 Self-correcting approximation techniqu 385 Semi-Markov process, 91 Serialization, 505 Service blocking, 548 rate, 266 load-dependent, 326, 369 station, 263 load-dependent, 352 single, 263 time distribution, 227 generally distributed, 295 remaining, 223 Shadow equation, 491 model, 479 network, 479 node, 479 service times, 479 technique, 478, 491 extended, 623 Shared cache blocks, 668 SHARPE, 500, 507, 594 Short circuit, 369 Short circuit, 452 Simultaneous displacement, 137 Simultaneous resource possession, 497, Single class closed network, 313 Single server node, 269 Single station queueing system, 209 Single-buffer polling system, 98 SIRO, 306 Slow state, 191 SMP, 91 Sojourn time, 48 Solution technique numerical, 277 SOR, 140, 594 Speedup, 521 Split-merge system, 519 Splitting, 441 SPN, 579 SPNP, 579 iSPN, 585 Squared coefficient of variation, 17-19 Standard deviation, 7 State diagram, 279 State probabilities, 251-252, 255, 277 State space, 36, 296 State transition diagram, 38, 251, 277 Stationary probability vector, 41

725

e,

501

726

INDEX

Stationary, 41 Steady-state probability, 253, 283 probability vector, 53, 274-275 solution, 278 Stochastic process, 36 Stochastic transition matrix, 38 Storage hierarchy, 566 Strategy last batch processor sharing (LBPS), WEIRDP, 307 Substitute network, 452 Success over-relaxation, 140 SUM, 397, 405, 534 Summation method, 397, 463 Surrogate delays, 504 Symmetric system GI/G/m, 249 M/M/m, 249 Synchronization overhead, 521 System availablity, 62 equation, 379 performance, 63 reliability, 63

Takahashi’s Method, 166 Tangible marking, 86 Task completion, 64 precedence graph, 517 Terminal system, 265, 513 Terminals, 265 Theorem arrival, 326, 343 BCMP, 300-301 Gordon/Newell, 292 Jackson’s, 632, 284 Little, 213 Little’s, 271 Norton’s, 368 PASTA, 343

Theorems

307

central limit, 24 Throughput, 213, 266, 270, 272 Time-average accumulated reward, 68 Time-homogeneous, 37 Timed transition, 86 Token rotation time, 614 Token, 84 TPM, 43 Traffic equations, 266, 286, 432, 614 Transfer blocking, 548 time, 502 Transient state, 47 Transient state probabilities, 41 Transient uniformization, 184 Transition, 84 enabled, 84 fire, 84 immediate, 86 probability, 49 rates, 50 timed, 86 Tree-convolution, 326

U Uniformization, Unit vector, UNIX, 620 Upper time Utilization,

104 53 limit, 212,

246 221, 269,

272

VW Vanishing marking, 86 Variabilities in workload, Variance, 7, 9, 11, 13-14, Visit ratio, 266, 289, 297 Wafer production system, Waiting time, 213 WAN,

610

XYZ z-transform,

25,

27

418 16-20, 639

214,

217